---
abstract: 'A simplicial cell ball is a simplicial poset whose geometric realization is homeomorphic to a ball. Recently, Samuel Kolins gave a series of necessary conditions and sufficient conditions on $h$-vectors of simplicial cell balls, and characterized them up to dimension $6$. In this paper, we extend Kolins’ results. We characterize all possible $h$-vectors of simplicial cell balls in arbitrary dimension.'
address: ' Satoshi Murai, Department of Mathematical Science, Faculty of Science, Yamaguchi University, 1677-1 Yoshida, Yamaguchi 753-8512, Japan. '
author:
- Satoshi Murai
title: '$h$-vectors of simplicial cell balls'
---

Introduction
============

A *simplicial poset* is a finite poset $P$ with a minimal element $\hat 0$ such that every interval $[\hat 0, \sigma]$ for $\sigma \in P$ is a Boolean algebra. A typical example of a simplicial poset is the face poset of a simplicial complex, but not all simplicial posets come from simplicial complexes. Simplicial posets are CW-posets. Thus for any simplicial poset $P$ there is a regular CW-complex $\Gamma(P)$ whose face poset is isomorphic to $P$ (see [@Bj pp. 8–9]). A simplicial poset $P$ (or a CW-complex $\Gamma(P)$) is called a *simplicial cell $d$-ball (respectively $d$-sphere)* if $\Gamma(P)$ is homeomorphic to a $d$-ball (respectively $d$-sphere). Kolins [@Ko] studied $h$-vectors of simplicial cell balls and gave a characterization of $h$-vectors of simplicial cell balls up to dimension $6$. In this paper, we extend the results of Kolins and give a complete characterization of $h$-vectors of simplicial cell balls.

Let $P$ be a simplicial poset. We say that an element $\sigma \in P$ has *rank $i$*, denoted ${\ensuremath{\mathrm{rank}}\hspace{1pt}}\sigma =i$, if $[\hat 0,\sigma]$ is a Boolean algebra of rank $i$. The *dimension* of $P$ is $$\dim P= \max \{{\ensuremath{\mathrm{rank}}\hspace{1pt}}\sigma : \sigma \in P\}-1.$$ Let $d=\dim P +1$ and let $f_i=f_i(P)$ be the number of elements $\sigma \in P$ having rank $i+1$ for $i=-1,0,\dots,d-1$. Thus $f_i$ is the number of $i$-dimensional cells of $\Gamma (P)$. The vector $f(P)=(f_{-1},f_0,\dots,f_{d-1})$ is called the *$f$-vector (face vector) of $P$*. We define the *$h$-vector $h(P)=(h_0(P),h_1(P),\dots,h_d(P)) \in { \ensuremath{\mathbb{Z}}}^{d+1}$ of $P$* by the relation $$\sum_{i=0}^d f_{i-1}  (1-t)^{d-i} = \sum_{i=0}^d h_i(P) t^{d-i}.$$ Then knowing $f(P)$ is equivalent to knowing $h(P)$.

On face vectors of simplicial cell spheres, Stanley [@St] and Masuda [@Ma] proved the following result, which characterizes all possible $h$-vectors of simplicial cell spheres.

\[1.1\] Let ${\mathbf{h}}=(h_0,h_1,\dots,h_d) \in { \ensuremath{\mathbb{Z}}}^{d+1}$. Then ${\mathbf{h}}$ is the $h$-vector of a simplicial cell $(d-1)$-sphere if and only if it satisfies the following conditions:

-   $h_0=h_d=1$ and $h_i=h_{d-i}$ for $i=1,2,\dots,d-1$.

-   $h_i \geq 0$ for $i=0,1,\dots,d$.

-   If $h_n=0$ for some $1 \leq n \leq d-1$ then $\sum_{k=0}^d h_k$ is even.

For a vector ${\mathbf{h}}=(h_0,h_1,\dots,h_d) \in { \ensuremath{\mathbb{Z}}}^{d+1}$, we define $\partial {\mathbf{h}}=(\partial h_0,\partial h_1,\dots,\partial h_{d-1}) \in { \ensuremath{\mathbb{Z}}}^d$ by $$\partial h_i = (h_0+h_1+\cdots +h_i)-(h_d+h_{d-1}+\cdots +h_{d-i})$$ for $i=0,1,\dots,d-1$. It is known that if ${\mathbf{h}}$ is the $h$-vector of a simplicial cell $(d-1)$-ball $P$, then $\partial {\mathbf{h}}$ is the $h$-vector of the boundary sphere of $P$. In this paper, we prove the next result, which characterizes all possible $h$-vectors of simplicial cell balls.

\[1.2\] Let ${\mathbf{h}}=(h_0,h_1,\dots,h_d) \in { \ensuremath{\mathbb{Z}}}^{d+1}$. Then ${\mathbf{h}}$ is the $h$-vector of a simplicial cell $(d-1)$-ball if and only if it satisfies the following conditions:

-   $h_0=1,$ $h_d=0$ and $h_k \geq 0$ for $k=1,2,\dots,d-1$.

-   $\partial h_k \geq 0$ for $k=0,1,\dots,d-1$.

-   If $d$ is odd and $\partial h_n=0$ for some $1 \leq n \leq d-2$ then $\sum_{k=0}^d h_k$ is even.

-   If $\partial h_n=0$ for some $1 \leq n \leq d-2$ then $$h_k + h_{k-1} + \cdots +h_{k-n+1} \geq \partial h_k
    \ \ \mbox{ for } k=n,n+1,\dots,d-1.$$

-   If $\partial h_i=0$ and $h_j=0$ for some positive integers $i$ and $j$ with $i+j \leq d$ then $\sum_{k=0}^d h_k$ is even.

-   Suppose $\partial h_n = 0$ for some $1 \leq n < \frac d 2$. If $(h_\ell + h_{\ell-1} + \cdots + h_{\ell-n+1}) -\partial h_\ell < n$ for some $ n \leq \ell \leq d-n$ then $\sum_{k=0}^d h_k$ is even.

-   Suppose $\partial h_i=0$ and $h_j=0$ for some integers $i$ and $j$ with $0< i < \frac d 2$ and $d-i  < j <d$. If $\partial h_\ell \leq \ell$ for some $\ell \leq d-j$ then $\sum_{k=0}^d h_k$ is even.

Note that conditions (5), (6) and (7) are unnecessary when $d$ is odd. We also note that some special cases of the above theorem are due to Kolins [@Ko]. For necessity, (1), (2), (3) and some special cases of (4), (5) and (7) (the condition (4) for $n=1$, the condition (5) for $i=1,2$ and $j=1$, and the condition (7) for $\ell=1$) were proved in [@Ko Sections 3, 4 and 5]. Also, when $\partial {\mathbf{h}}$ is positive or $\sum_{k=0}^d h_k$ is even, the sufficiency of the theorem was proved in [@Ko Theorem 16]. In particular, for even dimensional simplicial cell balls, the sufficiency of the theorem is due to Kolins.

A simplicial poset $P$ (or a CW-complex $\Gamma(P)$) is called a *simplicial cell decomposition of a topological space $X$* if $\Gamma(P)$ is homeomorphic to $X$. It was proved in [@Mu] that face vectors of simplicial cell decompositions of some manifolds without boundary, such as product of spheres and real projective spaces, are characterized by conditions similar to those in Theorem \[1.1\]. It would be of interest to find manifolds $M$ with boundary such that face vectors of simplicial cell decompositions of $M$ can be characterized by conditions similar to those in Theorem \[1.2\].

This paper is organized as follows. In Section 2, we prove the necessity of Theorem \[1.2\] by using the face ring of a simplicial poset. In Section 3, we prove the sufficiency of Theorem \[1.2\] by using constructibility and shellability.

Proof of necessity
==================

In this section, we prove the necessity of Theorem \[1.2\]. To prove this, we need the face ring of a simplicial poset introduced by Stanley [@St]. We assume familiarity with commutative algebra theory.

We recall some basic notations on commutative algebra. Let $K$ be an infinite field, $R=R_0 \bigoplus R_1 \bigoplus \cdots$ a finitely generated commutative graded $K$-algebra with $R_0=K$, where $R_k$ is the graded component of $R$ of degree $k$, and let $M$ be a finitely generated graded $R$-module. The *Hilbert series $H_M(t)$ of $M$* is the formal power series $H_M(t)=\sum _{k = 0}^\infty (\dim_K M_k )t^k$. The *Krull dimension* of $M$, denoted $\dim M$, is the minimal $m$ for which there exist homogeneous elements $\theta_1,\dots,\theta_m \in R$ of positive degrees such that $M/(\theta_1,\dots,\theta_m)M$ is a finite dimensional $K$-vector space. When $\dim M=d$, a sequence of homogeneous elements $\theta_1,\dots,\theta_d \in R$ of positive degrees such that $M/(\theta_1,\dots,\theta_d)M$ is a finite dimensional $K$-vector space is called a *homogeneous system of parameters* (*h.s.o.p. *for short) *of* $M$. Moreover, if $\theta_1,\dots,\theta_d$ have degree $1$, then we call $\theta_1,\dots,\theta_d$ a *linear system of parameters* (*l.s.o.p. *for short) *of* $M$. We say that $M$ is *Cohen-Macaulay* if, for every (equivalently some) h.s.o.p. $\theta_1,\dots,\theta_d$ of $M$, one has that $\theta_i$ is a non-zero divisor of $M/(\theta_1,\dots,\theta_{i-1})M$ for all $i=1,2,\dots,d$. Also, $R$ is said to be Cohen-Macaulay if it is a Cohen-Macaulay $R$-module.

Let $P$ be a $(d-1)$-dimensional simplicial poset and let $A=K[x_\sigma: \sigma \in P \setminus \{ \hat 0\}]$ be the polynomial ring over an infinite field $K$ in indeterminates indexed by the elements in $P \setminus \{\hat 0\}$. We define the grading of $A$ by $\deg x_\sigma = {\ensuremath{\mathrm{rank}}\hspace{1pt}}\sigma$. The *face ring of $P$* is the quotient ring $K[P]=A/I_P$, where $I_P$ is the ideal generated by the following elements:

-   $x_\sigma x_\tau$, if $\sigma,\tau \in P$ have no common upper bounds in $P$.

-   $x_\sigma x_\tau - x_{\sigma \wedge \tau} \sum_\rho x_\rho$, where the summation runs over the all minimal upper bounds of $\sigma$ and $\tau$ and where $\sigma \wedge \tau$ is the meet (largest lower bound) of $\sigma$ and $\tau$, otherwise. (We consider $x_{\sigma \wedge \tau}=1$ if $\sigma \wedge \tau=\hat 0$.)

Note that, for all $\sigma,\tau \in P$ such that $\sigma$ and $\tau$ have a common upper bound $\rho$, there is the unique largest lower bound of $\sigma$ and $\tau$ since $[\hat 0,\rho]$ is a Boolean algebra. Also, since generators of $I_P$ are homogeneous, the ring $K[P]$ is graded. In the special case when $P$ is the face poset of a simplicial complex, the ring $K[P]$ is isomorphic to the face ring (Stanley-Reisner ring) of a simplicial complex [@St2 p. 53]. See [@MMP p. 467].

A simplicial poset $P$ is said to be *Cohen-Macaulay* if the ring $K[P]$ is Cohen-Macaulay. Note that this is equivalent to saying that the order complex of $P \setminus \{\hat 0\}$ is a Cohen-Macaulay simplicial complex. On face rings of simplicial posets, the following properties are known.

-   The Krull dimension of $K[P]$ is $\dim P +1$.

-   $K[P]$ has an l.s.o.p.

-   $H_{K[P]}(t)=(h_0+h_1t+ \cdots + h_dt^d) /(1-t)^d$, where $d=\dim P +1$ and where $(h_0,h_1,\dots,h_d)=h(P)$.

-   If $\Gamma(P)$ is homeomorphic to a ball or a sphere then $P$ is Cohen-Macaulay.

See [@St pp. 325–326] for the first three properties. The last property follows from a topological criterion of Cohen-Macaulay simplicial complexes [@St2 II, Corollary 4.2 and Proposition 4.3].

A subset $I$ of a simplicial poset $P$ is said to be an *order ideal of* $P$ if $\sigma \in I$ and $\tau \leq \sigma$ imply $\tau \in I$. Thus an order ideal of a simplicial poset is again a simplicial poset. For a simplicial poset $P$ and elements $\sigma_1,\dots,\sigma_k \in P$, we write $\langle\sigma_1,\dots,\sigma_k \rangle$ for the order ideal of $P$ generated by $\sigma_1,\dots,\sigma_k$, in other words, $$\langle \sigma_1,\dots,\sigma_k\rangle=\{\tau \in P: \mbox{ there is $1 \leq j \leq k$ such that } \tau \leq \sigma_j\}.$$ For elements $f_1,\dots,f_m$ in a ring $A$, we write $(f_1,\dots,f_m)$ for the ideal of $A$ generated by $f_1,\dots,f_m$. We often use the following obvious fact.

\[subposet\] Let $P$ be a simplicial poset and $A=K[x_\sigma: \sigma \in P \setminus \{\hat 0\}]$. If $Q$ is an order ideal of $P$ then $A/(I_P+(x_\sigma: \sigma \not \in Q))$ is isomorphic to $K[Q]$ as a ring.

Let $P$ be a simplicial cell $(d-1)$-ball. Then each rank $d-1$ element of $P$ is covered by at most two elements. The *boundary $\partial P$ of $P$* is the order ideal $$\partial P=\langle \sigma \in P: {\ensuremath{\mathrm{rank}}\hspace{1pt}}\sigma=d-1,\ \sigma \mbox{ is covered by exactly one element in }P\rangle.$$ Since $\partial P$ is the face poset of the boundary cell complex of $\Gamma(P)$, $\partial P$ is a simplicial cell $(d-2)$-sphere. Moreover, it is known that if ${\mathbf{h}}=(h_0,h_1,\dots,h_d)$ is the $h$-vector of $P$ then $\partial {\mathbf{h}}=(\partial h_0,\partial h_1,\dots,\partial h_{d-1})$ is the $h$-vector of $\partial P$. See [@Ko Section 3]. On $h$-vectors of simplicial cell balls, the following result was proved in [@Ko Theorem 5].

\[kolins\] Let $P$ be a simplicial cell $(d-1)$-ball and ${\mathbf{h}}=h(P)=(h_0,h_1,\dots,h_d)$. Then

-   $h_0=1,$ $h_d=0$ and $h_k \geq 0$ for $k=1,2,\dots,d-1$.

-   $\partial h_k \geq 0$ for $k=0,1,\dots,d-1$.

-   If $d$ is odd and $\partial h_n=0$ for some $1 \leq n \leq d-2$ then $\sum_{k=0}^d h_k$ is even.

We sketch a proof of the above theorem. The vectors ${\mathbf{h}}$ and $\partial {\mathbf{h}}$ are non-negative since $h$-vectors of Cohen-Macaulay simplicial posets are non-negative [@St Theorem 3.10]. Also, since $h_d(P)$ is equal to the reduced Euler characteristic of $P$ times $(-1)^{d-1}$, $h_d(P)=0$ if $P$ is a simplicial cell ball. Finally, (3) follows from Theorem \[1.1\] as follows: If $d$ is odd then $$\partial h_{\frac {d-1} 2}= (h_0+\cdots +h_{\frac {d-1} 2}) - (h_{\frac {d+1} 2}+ \cdots +h_d)$$ is equal to $\sum_{k=0}^d h_k$ mod $2$. Since $\sum_{k=0}^{d-1} \partial h_k$ is even by Theorem \[1.1\](3), by the symmetry $\partial h_i=\partial h_{d-1-i}$ of $\partial {\mathbf{h}}$, $\partial h_{\frac {d-1} 2}$ must be even.

In the rest of this section, we prove that conditions (4), (5), (6) and (7) in Theorem \[1.2\] are necessary conditions of $h$-vectors of simplicial cell balls.

Proof of (4)
------------

\[trivial\] Let $P$ be a $(d-1)$-dimensional simplicial poset and $h(P)=(h_0,\dots,h_d)$. Let $K[P]=A/I_P$ be the face ring of $P$ and $\theta_1,\dots,\theta_d \in A_1$ an l.s.o.p. of $K[P]$. If $K[P]$ is Cohen-Macaulay then, for all integers $n >0$ and $n \leq k \leq d$, one has $$\dim_K \big(A/\big(I_P+(\theta_1,\dots,\theta_{d-1},\theta_d^n)\big)\big)_k
=h_k+h_{k-1} + \cdots + h_{k-n+1}.$$

Let $R[i]=A/(I_P+(\theta_1,\dots,\theta_{i}))$ for $i=0,1,\dots,d-1$. Since $R[0]=K[P]$ is Cohen-Macaulay, we have the exact sequence $$0 \longrightarrow R[i-1] \stackrel{\times \theta_i}{\longrightarrow} R[i-1] \longrightarrow R[i] \longrightarrow 0$$ for $i=1,2,\dots,d-1$. The exact sequences show that $H_{R[i]}(t)=(1-t)H_{R[i-1]}(t)$ for $i=1,2,\dots,d-1$. Then, since $H_{K[P]}(t)=(h_0+h_1t+\cdots+h_dt^d)/(1-t)^d$, we have $H_{R[d-1]}(t)=(h_0+h_1t+\cdots+h_dt^d)/(1-t)$. This fact says $$\dim_K \big(R[d-1]\big)_k = h_k + h_{k-1} + \cdots + h_1 +h_0$$ for $k=0,1,\dots,d.$ Also, since $\theta_1,\dots,\theta_{d-1},\theta_d^n$ is an h.s.o.p. of $K[P]$, the sequence $$0 \longrightarrow R[d-1] \stackrel{\times \theta_d^n}{\longrightarrow} R[d-1] \longrightarrow A/\big(I_P+(\theta_1,\dots,\theta_{d-1},\theta_d^n)\big) \longrightarrow     0$$ is exact. Hence $$\begin{aligned}
\dim_K \big(A/\big(I_P+(\theta_1,\dots,\theta_{d-1},\theta_d^n)\big)\big)_k
&=& \dim_K \big(R[d-1] \big)_k - \dim_K \big( R[d-1] \big)_{k-n}\\
&=& h_k+h_{k-1} + \cdots + h_{k-n+1}\end{aligned}$$ for $k=n,n+1,\dots,d$, as desired.

\[necessary1\] Let $P$ be a simplicial cell $(d-1)$-ball and ${\mathbf{h}}=h(P)=(h_0,h_1,\dots,h_d)$. If $\partial h_n=0$ for some $1 \leq n \leq d-2$ then $$h_k + h_{k-1} + \cdots +h_{k-n+1} \geq \partial h_k
\ \ \mbox{ for } k=n,n+1,\dots,d-1.$$

Let $A=K[x_\sigma: \sigma \in P \setminus \{ \hat 0\}]$ and $J=I_P + (x_\sigma: \sigma \not \in \partial P) \subset A$. Then $A/J \cong K[\partial P]$ by Lemma \[subposet\]. For a general choice of linear forms $\theta_1,\dots,\theta_d \in A_1$, we have (e.g. use a criterion of an l.s.o.p. for face rings [@Ko Lemma 6])

-   $\theta_1,\dots,\theta_{d-1}$ is an l.s.o.p. of $A/J$.

-   $\theta_1,\dots,\theta_{d-1},\theta_d$ is an l.s.o.p. of $A/I_P$.

Let $\overline \Theta = \theta_1,\dots,\theta_{d-1}$. Since $A/J$ is a $(d-1)$-dimensional Cohen–Macaulay ring, $$\begin{aligned}
\label{B}
\dim_K \left(A/ \left( J+ (\overline \Theta) \right) \right)_k = h_k(\partial P)=\partial h_k\end{aligned}$$ for $k = 0,1,\dots,d-1$. In particular, since $\dim_K(A/(J+(\overline{\Theta})))_n=\partial h_n =0$ by the assumption, we have $\theta_d^n \in J+ (\overline {\Theta}).$ Then, since $J \supset I_P$, we have $J+(\overline\Theta) \supset I_P + (\overline\Theta)+(\theta_d^n)$. Thus $$\begin{aligned}
\dim_K \big( A/ \big( I_P + (\overline {\Theta})+ (\theta_d^n ) \big) \big)_ k \geq \dim_K \left( A/ \left( J+(\overline {\Theta}) \right) \right)_k=\partial h_k\end{aligned}$$ for all $k =0,1,\dots,d-1$. Then the statement follows from Lemma \[trivial\].

Proof of (5), (6) and (7)
-------------------------

Let $P$ be a $(d-1)$-dimensional simplicial poset. A rank $1$ element of $P$ is called a *vertex of $P$* and a maximal element (w.r.t. a partial order) of $P$ is called a *facet of $P$*. The *cone of $P$* is the product of the posets $$C(P)=P \times \{1,2\}=\{(\sigma,i): \sigma \in P,\ i=1,2\},$$ where the ordering is defined by $(\sigma,i) \geq (\tau,j)$ if $\sigma \geq \tau$ and $i \geq j$. Then $C(P)$ is simplicial and $\Gamma(C(P))$ is homeomorphic to the (topological) cone of $\Gamma(P)$. Also, a straightforward computation shows $h_i(P)=h_i(C(P))$ for all $i=0,1,\dots,d$ and $h_{d+1}(C(P))=0$.

Let $P$ be a simplicial cell ball. We define the simplicial poset $SP$ by $$SP= P \cup C(\partial P)/\! \sim$$ where $\sim$ is the equivalence relation defined by $\sigma \sim (\sigma,1)$ for all $\sigma \in \partial P$. In other words, $SP$ is a simplicial cell sphere such that $\Gamma(SP)$ is obtained from $\Gamma(P)$ by coning off its boundary.

We need the next result due to Masuda [@Ma Section 5] (see also [@MR] for a simplified proof).

\[2-2\] Let $P$ be a simplicial cell $(d-1)$-sphere, $K[P]=A/I_P$ the face ring of $P$ and $\Theta=\theta_1,\dots,\theta_d \in A_1$ an l.s.o.p. of $K[P]$. If $x_{v_1} x_{v_2} \cdots x_{v_d}$ is zero in $A/(I_P+(\Theta))$ for any sequence of $d$ distinct vertices $v_1,v_2,\dots,v_d$ of $P$ then the number of the facets of $P$ is even.

In the rest of this section, we fix the following notation: Let $P$ be a simplicial cell $(d-1)$-ball and $${\mathbf{h}}=h(P)=(h_0,h_1,\dots,h_d).$$ Let $Q=C(\partial P)$ and $SP=P \cup C(\partial P)/\! \sim$ be simplicial posets defined as above. Also, let $A=K[x_\sigma: \sigma \in SP \setminus\{\hat 0\}]$, $$J=I_{SP}+(x_\sigma: \sigma \in SP \setminus P)=I_{SP}+(x_\sigma: \sigma \in Q \setminus \partial P),$$ $$L= I_{SP}+(x_\sigma: \sigma \in SP \setminus Q)
=I_{SP}+(x_\sigma: \sigma \in P \setminus \partial P),$$ and let $\Theta=\theta_1,\theta_2,\dots,\theta_d \in A_1$ be a sequence of general linear forms. Then we have the following properties.

-   $A/J \cong K[P]$, $A/L\cong K[Q]$ and $$A/(L+J)= A/\big(I_{SP} + (x_\sigma: \sigma \in SP \setminus \partial P)\big) \cong K[\partial P].$$

-   $\overline \Theta =\theta_1,\dots,\theta_{d-1}$ is an l.s.o.p. of $A/(L+J)$ and $\Theta=\theta_1,\dots,\theta_{d}$ is a common l.s.o.p. of $A/I_{SP}$, $A/J$ and $A/L$.

Recall that $\sum_{k=0}^d h_k$ is equal to the number of the facets of $P$. The next lemma shows that, to prove (5)–(7), it is enough to study the number of the facets of $SP$.

\[facets\] Suppose $\partial h_i=0$ for some $1 \leq i \leq d-2$. If the number of the facets of $SP$ is even then $\sum_{k=0}^d h_k$ is even.

By the construction of $SP$, the number of the facets of $P$ is equal to that of $SP$ minus that of $\partial P$. Then the statement follows from Theorem \[1.1\](3).

\[int1\] Let $v_1,v_2,\dots,v_d$ be distinct vertices of $SP$. Then

-   If $\partial h_i=0$ for some $1 \leq i \leq d-2$ then $x_{v_1}x_{v_2} \cdots x_{v_i} \in L+(\Theta)$.

-   If $h_j=0$ for some $1 \leq j \leq d$ then $x_{v_d} x_{v_{d-1}} \cdots x_{v_{d-j+1}} \in
    J+(\Theta)$.

Recall $A/J \cong K[P]$ and $A/L \cong K[Q]$. Since $Q$ is the cone of $\partial P$, $$\begin{aligned}
\label{XX1}
\dim_K \big( A/ \big( L+(\Theta) \big) \big)_k=h_k(Q)=h_k(\partial P)=\partial h_k\end{aligned}$$ for $k =0,1,\dots,d-1$. Thus, $\partial h_i=0$ implies $x_{v_{1}}x_{v_2} \cdots x_{v_i} \in L+(\Theta).$ Similarly, since $\dim_K (A/(J+(\Theta)))_j=h_j$, $h_j=0$ implies $x_{v_d} x_{v_{d-1}} \cdots x_{v_{d-j+1}} \in J+(\Theta)$.

Now we prove (5) and (6).

\[2-3\] If $\partial h_i=0$ and $h_j=0$ for some positive integers $i$ and $j$ with $i+j \leq d$ then $\sum_{k=0}^d h_k$ is even.

Let $v_1,v_2,\dots,v_d$ be distinct vertices of $SP$. By Lemmas \[2-2\] and \[facets\], it is enough to prove that the monomial $x_{v_1}x_{v_2}\cdots x_{v_d}$ is zero in $A/(I_{SP} +(\Theta))$.

Since $i+j \leq d$, Lemma \[int1\] shows $$x_{v_1} x_{v_2} \cdots x_{v_d} \in
LJ+(\Theta) \subset
 I_{SP} +(\Theta)+(x_\sigma x_\tau :
\sigma \in P \setminus \partial P,\
\tau \in Q \setminus \partial P).$$ However, for all $\sigma \in P \setminus \partial P$ and $\tau \in Q \setminus \partial P$, we have $x_\sigma x_\tau \in I_{SP}$ since $\sigma$ and $\tau$ have no common upper bounds. Thus $x_{v_1} x_{v_2} \cdots x_{v_d} \in I_{SP}+(\Theta)$.

\[necessary7\] Suppose $\partial h_n = 0$ for some $1 \leq n < \frac d 2$. If $(h_\ell + h_{\ell-1} + \cdots + h_{\ell-n+1}) -\partial h_\ell < n$ for some $ n \leq \ell \leq d-n$ then $\sum_{k=0}^d h_k$ is even.

Let $v_1,v_2,\dots,v_d$ be distinct vertices of $SP$. Then $x_{v_1} x_{v_2} \cdots x_{v_n} \in L+(\Theta)$ by Lemma \[int1\](i). Since $n+ \ell \leq d$, by the same argument as in the proof of Theorem \[2-3\], it is enough to prove that $$x_{v_d}x_{v_{d-1}}\cdots x_{v_{d-\ell+1}} \in J+(\Theta).$$

Consider the exact sequence $$0 \longrightarrow
N\longrightarrow
A/\big(J+(\overline \Theta,\theta_d^n)\big)
\longrightarrow
A/\big(L+J+(\overline \Theta,\theta_d^n)\big)
\longrightarrow 
0,$$ where $N=(L+J+(\overline \Theta,\theta_d^n))
/(J+(\overline \Theta,\theta_d^n))$. Since $A/(L+J)\cong K[\partial P]$ and since $\overline \Theta$ is an l.s.o.p. of $A/(L+J)$, we have $\dim_K (A/(L+J+(\overline \Theta)))_n=\partial h_n=0$ and $$A_n=\big(L+J+(\overline \Theta)\big)_n.$$ In particular, we have $L+J+(\overline \Theta)=L+J+(\overline \Theta,\theta_d^n)$. Then, by Lemma \[trivial\] we have $$\begin{aligned}
\label{qqq}
\hspace{16pt} \dim_K N_\ell&=&\dim_K \big(A/\big(J+(\overline \Theta,\theta_d^n)\big)
\big)_\ell
-
\dim_K \big(A/\big(L+J+(\overline \Theta,\theta_d^n)\big)
\big)_\ell\\
\nonumber & =&
(h_\ell+ h_{\ell-1}+\cdots + h_{\ell-n+1}) - \partial h_\ell \\
\nonumber &\leq& n-1,\end{aligned}$$ where the last inequality follows from the assumption.

Let $u_k=x_{v_d} x_{v_{d-1}} \cdots x_{v_{d-k+1}}$ for $k=1,2,\dots,\ell$. Since $A_n=(L+J+(\overline \Theta))_n$, any element in $A$ of degree $n$ is contained in $L+J+(\overline \Theta)$. Since $\ell \geq n$, this fact says that the elements $$u_\ell,u_{\ell-1} \theta_d,\dots,u_{\ell-n+1}\theta_d^{n-1}$$ are contained in $L+J+(\overline \Theta)$ since they are products of elements of degree $1$. Then says that the above $n$ elements are $K$-linearly dependent in $N$. Thus there are $\alpha_0,\alpha_1,\dots,\alpha_{n-1} \in K$ with $(\alpha_0,\alpha_1,\dots,\alpha_{n-1}) \ne (0,0,\dots,0)$ such that $$\alpha_0 u_\ell + \alpha_1 u_{\ell-1} \theta_d + \cdots + \alpha_{n-1} u_{\ell-n+1}\theta_d^{n-1}
\in J+(\overline \Theta,\theta_d^n).$$ Let $s = \min \{k : \alpha_k \ne 0\}$. Then there is an $h \in A$ such that $$\theta_d^s \alpha_s u_{\ell-s} + \theta_d^{s+1} h \in J+ (\overline \Theta).$$ Since $\Theta$ is an l.s.o.p. of a Cohen-Macaulay ring $A/J$, $\theta_d$ is a non-zero divisor of $A/(J+(\overline \Theta))$. Thus $$\alpha_s u_{\ell-s}  + \theta_d h \in  J +(\overline \Theta).$$ This fact implies $u_{\ell-s} \in J+(\Theta)$. Since $u_{\ell-s}$ divides $u_\ell$, we have $$x_{v_d}x_{v_{d-1}} \cdots x_{v_{d-\ell+1}}
= u_{\ell}
\in J+(\Theta),$$ as desired.

Next, we study the condition (7). To simplify the notation, we write $R=A/J \cong K[P]$ and let $$C=(L+J)/J=
\big((x_\sigma: \sigma \in P \setminus \partial P)+J \big)/J.$$ Thus $C$ is the ideal of $R$ generated by the interior faces of $P$.

\[2.3.1\] $C$ is a $d$-dimensional Cohen-Macaulay $A$-module such that $H_C(t)={(h_d+h_{d-1}t + \cdots + h_0t^d)} /{(1-t)^d}$.

Consider the exact sequence $$\begin{aligned}
\label{E2.4}
0 \longrightarrow
C \longrightarrow
R \longrightarrow
R/C \longrightarrow
0.\end{aligned}$$ Observe $R \cong K[P]$ and $R/C = A/(L+J) \cong K[\partial P]$. Then $$H_C(t)=H_R(t)-H_{R/C}(t)=H_{K[P]}(t)-H_{K[\partial P]}(t)
=\frac {(h_d+h_{d-1}t + \cdots + h_0t^d)} {(1-t)^d}.$$ Also, $C$ is a $d$-dimensional Cohen-Macaulay $A$-module since $R/C$ is a $(d-1)$-dimensional Cohen-Macaulay $A$-module and $R$ is a $d$-dimensional Cohen-Macaulay $A$-module (e.g., use [@E Corollary 18.6]).

The ideal $C$ is the canonical module of $R$ (see [@BH Chapter 3]). This fact can be proved in the same way as in the proof of [@BH Theorem 5.7.1], and gives an alternative proof of Lemma \[2.3.1\].

Since $C$ is a submodule of $R$ with $\dim C=\dim R$ and since $C$ is Cohen-Macaulay, $\Theta$ is an l.s.o.p. of $C$ and we have $$\dim_K \big(C/(\Theta C)\big)_i=h_{d-i}$$ for $i=0,1,\dots,d$ by Lemma \[2.3.1\]. This fact shows

\[E2.5\] If $h_j=0$ for some $1 \leq j \leq d-1$ then $C_{d-j}=\Theta C_{d-j-1}.$

\[2.3.2\] Suppose $h_j=0$ for some $1 \leq j \leq d-1$. If $\theta_d f  \in (C+\overline \Theta R)_{d-j}$ then, for any linear form $l \in R_1$, one has $l f \in C+\overline \Theta R$.

By Lemma \[E2.5\], $\theta_d f \in \overline \Theta R + \theta_d C$. Thus there is $g \in C$ such that $\theta_d (f-g) \in \overline \Theta R$. Since $\theta_d$ is a non-zero divisor of $R/(\overline \Theta R)$, we have $f-g \in \overline \Theta R$ and $f \in C + \overline \Theta R$. Hence $l f \in C+\overline \Theta R$.

The next theorem completes the proof of the necessity of Theorem \[1.2\].

\[necessity6\] Suppose $\partial h_i=0$ and $h_j=0$ for some integers $i$ and $j$ with $0< i < \frac d 2$ and $d-i  < j <d$. If $\partial h_\ell \leq \ell$ for some $\ell \leq d-j$ then $\sum_{k=0}^d h_k$ is even.

Let $v_1,v_2,\dots,v_d$ be distinct vertices of $SP$. Since $i<j$, in the same way as in the proof of Theorem \[2-3\], it is enough to prove that $$x_{v_d} x_{v_{d-1}}\cdots x_{v_{j+1}} \in J + (\Theta).$$

Let $u_k=x_{v_d} x_{v_{d-1}} \cdots x_{v_{d-k+1}}$ for $k=0,1,\dots,d-j$, where $u_0=1$. We regard each $u_k$ as an element in $R=A/J$. Recall that $\ell$ is an integer such that $\ell \leq d-j$ and $\partial h_\ell \leq \ell$. Since $\overline \Theta$ is an l.s.o.p. of $R/C=A/(L+J) \cong K[\partial P]$, $$\dim_K \big(R/(C+\overline \Theta R)\big)_\ell = \partial h_\ell \leq \ell.$$ Then the elements $$u_\ell, u_{\ell-1} \theta_d ,u_{\ell-2} \theta_d^2,\dots,u_0 \theta_d^\ell \in R_\ell$$ are $K$-linearly dependent in $R/(C+\overline \Theta R)$. Thus there are $\alpha_0,\alpha_1,\dots,\alpha_{\ell} \in K$ with $(\alpha_0,\alpha_1,\dots,\alpha_{\ell}) \ne 0$ such that $$\begin{aligned}
\alpha_0 u_\ell  + \alpha_1 u_{\ell-1} \theta_d + \cdots + \alpha_\ell u_0 \theta_d^\ell
\in 
C+\overline \Theta R.\end{aligned}$$ Let $s = \min\{ k : \alpha_k \ne 0\}$ and $h=(\sum_{k=s+1}^{\ell} \alpha_k u_{\ell-k} \theta_d^{k})/\theta_d^{s+1}$, where $h=0$ if $s=\ell$. Then $\theta_d^{s} (\alpha_{s} u_{\ell-s}+ \theta_d h ) \in (C+ \overline \Theta R)_\ell$ and $$\theta_d^{d-j-(\ell-s)} (\alpha_{s} u_{\ell-s}+ \theta_d h ) \in (C+ \overline \Theta R)_{d-j}.$$ By Lemma \[2.3.2\], we have $(u_{d-j}/u_{\ell-s}) (\alpha_s u_{\ell -s} + \theta_d h) \in C+\overline \Theta R$. Hence $$\alpha_s u_{d-j} \in (C+ \overline \Theta R+ \theta_d R)_{d-j} \subset \Theta R,$$ where the last inclusion follows from Lemma \[E2.5\]. Since $\alpha_s \ne 0$, $u_{d-j}$ is contained in $\Theta R$. This fact shows $u_{d-j}=x_{v_d} x_{v_{d-1}}\cdots x_{v_{j+1}} \in J + (\Theta)$ when we regard $u_{d-j}$ as an element of $A$.

Proof of sufficiency
====================

In this section, we prove the sufficiency of Theorem \[1.2\].

Since the proof is a bit long, we first give a short outline of the proof. The key tools of the proof are the two main lemmas outlined below.

-   In Lemma \[3.A\], we introduce a simple way to make a new simplicial cell ball from a given simplicial cell ball by gluing a pair of simplexes along their boundaries.

-   In Lemma \[3.B\], we construct a special simplicial cell ball whose boundary has a nice property.

Then we construct a simplicial cell ball with the desired $h$-vector as follows: First, starting from a simplex, we gradually make a larger simplicial cell ball by using (I) repeatedly; Then, at a certain point, we attach a simplicial cell ball in (II) to change the shape of the boundary; Finally, we make a simplicial cell ball with the desired $h$-vector by using (I) again. The first technique (I) appeared in [@Ko]. Indeed, if $\partial {\mathbf{h}}$ is positive or $\sum_{k=0}^d h_k$ is odd, then the first technique is enough to prove the desired statement. However, we need (II) when $\partial {\mathbf{h}}$ has a zero entry and $\sum_{k=0}^d h_k$ is even.

Before proving the main lemmas, we first study what happens to the $h$-vector when we glue two simplicial posets. Two finite posets $P$ and $Q$ are said to be *isomorphic* if there is a bijection $f:P \to Q$, called an *isomorphism*, such that, for all $ \sigma, \tau \in P$, $\sigma> \tau$ if and only if $f(\sigma) > f(\tau)$. We write $P \cong Q$ if $P$ and $Q$ are isomorphic.

Let $P$ and $Q$ be simplicial posets. Let $I \subset P$ and $J \subset Q$ be order ideals which are isomorphic as posets. For a given isomorphism $f : I \to J$, we define the equivalence relation $\sim_f$ on $P \cup Q$ by $\sigma \sim_f f(\sigma)$ for all $ \sigma \in I$, and define $$P \cup_f  Q = (P \cup Q) /\! \sim_f.$$ Then $P \cup_f Q$ is again a simplicial poset. We write $P \cup _I Q$ (or $P {{}_I \cup_J} Q$) for this simplicial poset $P \cup_f Q$ when the isomorphism $f$ is clear.

\[3.1\] Let $P$ and $Q$ be $(d-1)$-dimensional simplicial posets and let $I \subset P$ and $J \subset Q$ be order ideals with $I \cong J$. If $\dim I=d-2$ then $$h_k(P\cup_I Q)=h_k(P) + h_k(Q) -h_k(I) + h_{k-1}(I)$$ for $k=0,1,\dots,d$, where $h_{-1}(I)=h_d(I)=0$.

The desired formula follows from $f_k(P \cup_I Q)=f_k(P)+f_k(Q)-f_k(I)$ by straightforward computations.

For a positive integer $i$, we write $[i]=\{1,2,\dots,i\}$. For integers $1 \leq k \leq d$, let $$\Delta_d(k)=
\big\{F \subset [d]: F \not \supset [k]\big\}.$$ By defining a partial order on $\Delta_d(k)$ by inclusion, we regard $\Delta_d(k)$ as a simplicial poset of dimension $d-2$. In other words, $\Delta_d(k)$ is the (abstract) simplicial complex generated by $\{ [d] \setminus \{i\}: i=1,2,\dots,k\}$. Let ${\mathbf{e}}_0,{\mathbf{e}}_1,\dots,{\mathbf{e}}_d \in { \ensuremath{\mathbb{Z}}}^{d+1}$ be the unit vectors of ${ \ensuremath{\mathbb{Z}}}^{d+1}$. Thus ${\mathbf{e}}_i$ is the vector such that $(i+1)$-th entry of ${\mathbf{e}}_i$ is $1$ and all other entries of ${\mathbf{e}}_i$ are $0$. It is easy to see that $h_i(\Delta_d(k))=1$ for $i < k$ and $h_i(\Delta_d(k))=0$ for $i \geq k$. Then the next lemma follows from Lemma \[3.1\].

\[3.2\] With the same notation as in Lemma \[3.1\], if $I \cong \Delta_d(k)$ then $$h(P \cup_I Q)=h(P) + h(Q) - {\mathbf{e}}_0 + {\mathbf{e}}_{k}.$$

To apply Lemmas \[3.1\] and \[3.2\] efficiently, it is convenient to have a nice way to construct simplicial cell balls by gluing two simplicial cell balls. Constructibility and shellability give such nice ways.

A $(d-1)$-dimensional simplicial poset $P$ is said to be *constructible* if

-   $P$ is a Boolean algebra of rank $d$, or

-   there are $(d-1)$-dimensional constructible simplicial posets $Q$ and $Q'$ such that $P=Q { {}_I \cup_{I'} } Q'$ for some order ideals $I \subset Q$ and $I' \subset Q'$ and that $I \cong I'$ is a $(d-2)$-dimensional constructible simplicial poset.

A simplicial poset is said to be *pure* if all its facets have the same rank. A $(d-1)$-dimensional pure simplicial poset $P$ is said to be *shellable* if there is an order $\sigma_1,\sigma_2,\dots,\sigma_r$ of the facets of $P$, called a *shelling of $P$*, such that, for $i=2,3,\dots,r$, $\langle \sigma_1,\sigma_2,\dots,\sigma_{i-1} \rangle \cap \langle \sigma_i \rangle$ is isomorphic to $\Delta_d(k_i)$ for some $1 \leq k_i \leq d$.

Note that shellable simplicial posets are constructible since $\Delta_d(k)$ is constructible. In the rest of this paper, a simplicial cell $d$-ball which is constructible (respectively shellable) will be called a *constructible $d$-ball* (respectively *shellable $d$-ball*).

We need the following well-known properties of constructibility and shellability. See e.g., [@Bj2 Theorem 11.4 and Section 12].

-   Let $P$ be a simplicial cell sphere. If $Q \subsetneq P$ is a shellable simplicial poset with $\dim Q=\dim P$, then $Q$ is a simplicial cell ball.

-   Boolean algebras and $\Delta_d(k)$ with $k <d$ are shellable balls.

-   Let $P$ and $Q$ be constructible $d$-balls, and let $I \subsetneq \partial P$ and $J \subsetneq \partial Q$ be order ideals with $I \cong J$. If $I$ is a constructible $(d-1)$-ball then $P {_{I}\cup_J} Q$ is a constructible $d$-ball.

For a finite set $X$, we write $\# X$ for the cardinality of $X$. The next well-known lemma, which immediately follows from Lemma \[3.2\] and the fact that a Boolean algebra has the $h$-vector $(1,0,\dots,0)$, is useful to compute $h$-vectors of shellable simplicial posets.

\[shellablehvector\] Let $P$ be a $(d-1)$-dimensional shellable simplicial poset with a shelling $\sigma_1,\sigma_2,\dots,\sigma_r$. Then, for $i=1,2,\dots,d$, one has $$h_i(P)=\# \big\{ \ell \in \{2,3,\dots,r\}: 
\langle \sigma_1,\sigma_2,\dots,\sigma_{\ell-1} \rangle \cap \langle \sigma_\ell \rangle
\cong \Delta_d(i)
\big\}.$$

Now we prove our first main lemma. We say that a simplicial poset $P$ contains a simplicial poset $Q$ if there is an order ideal $I \subset P$ which is isomorphic to $Q$.

\[3.A\] Fix integers $1 \leq i \leq m \leq d$ and $1 \leq j \leq d-i$. Let $P$ be a constructible $(d-1)$-ball such that $\partial P$ contains $\Delta_d(m)$ and let $\ell = \min\{m,i+j\}$. Then there is a constructible $(d-1)$-ball $Q$ such that

-   $h(Q)=h(P) + {\mathbf{e}}_i + {\mathbf{e}}_{j}$, and

-   $\partial Q$ contains $\Delta_d(\ell)$.

By the assumption, there is an order ideal $L \subset \partial P$ which is isomorphic to $\Delta_d(m)$. We write $$L=\big\{\mathcal L(F): F \subset [d],\ F \not \supset [m]\big\},$$ where the ordering is defined by $\mathcal L(F)>\mathcal L(G)$ if $F \supset G$. Since $i \leq m$, $$I=\big\{\mathcal L(F): F \subset [d],\ F \not \supset [i]\big\} \cong \Delta_d(i)$$ is an order ideal of $\partial P$. We prepare two Boolean algebras $$A=\big\{{\mathcal{A}}(F): F \subset [d]\big\} \mbox{ and } B=\big\{{\mathcal{B}}(F): F \subset [d] \big\},$$ where the ordering is defined by inclusion on $F$. Consider the order ideal $$J=\big\{ {\mathcal{A}}(F): F \subset [d],\ F \not \supset \{i+1,\dots,i+j\}\big\}
\cong \Delta_d(j).$$ Let $$Q=
(P \cup_I A) \cup_J B
=(P \cup A \cup B) /\! \sim,$$ where $\sim$ is the equivalence relation defined by $\mathcal L(F) \sim {\mathcal{A}}(F)$ if $F \not \supset [i]$ and ${\mathcal{A}}(F) \sim {\mathcal{B}}(F)$ if $F \not \supset \{i+1,\dots,i+j\}$. Then $Q$ is a constructible $(d-1)$-ball and has the desired $h$-vector by Lemma \[3.2\].

It remains to prove that $\partial Q$ contains $\Delta_d(\ell)$. Clearly, $\partial Q$ contains an order ideal $$\Delta
= \big\{{\mathcal{B}}(F): F \subset [d],\ F \not \supset [i] \big\}
\bigcup \big\{\mathcal L(F): F \subset [d],\ F \not \supset \{i+1,\dots,\ell\}\big\}/\!\sim.$$ We claim that $\Delta \cong \Delta_d(\ell)$. What we must prove is that if $F \not \supset [i]$ and $F \not \supset \{i+1,\dots,\ell\}$ then $\mathcal L(F) \sim {\mathcal{B}}(F)$. Indeed, $F \not \supset [i]$ implies $\mathcal L (F) \sim {\mathcal{A}}(F)$. Also, since $\ell \leq i+j$, $F \not \supset \{i+1,\dots,\ell\}$ implies ${\mathcal{A}}(F)\sim {\mathcal{B}}(F)$.

We note that Lemma \[3.A\] essentially appeared in the proof of [@Ko Theorem 16].

Next, we develop a way to construct simplicial cell balls by applying Lemma \[3.A\] repeatedly. Let ${\mathbf{h}}=(h_0,h_1,\dots,h_d) \in { \ensuremath{\mathbb{Z}}}_{\geq 0}^{d+1}$ be a vector such that $\sum_{k=0}^d h_k$ is even. We write ${\mathbf{h}}={\mathbf{e}}_{i_1}+{\mathbf{e}}_{i_2}+ \cdots + {\mathbf{e}}_{i_a}$, where $a=\sum_{k=0}^d h_k$ and where $i_1 \leq i_2 \leq \cdots \leq i_a$. Consider the following decomposition of ${\mathbf{h}}$: $${\mathbf{h}}= ({\mathbf{e}}_{i_1} + {\mathbf{e}}_{i_a}) + ({\mathbf{e}}_{i_2} + {\mathbf{e}}_{i_{a-1}}) + \cdots + ({\mathbf{e}}_{i_{\frac a 2}}+{\mathbf{e}}_{i_{\frac a 2 +1}}).$$ We define the *initial number ${\ensuremath{\mathrm{in}}\hspace{1pt}}({\mathbf{h}})$ of ${\mathbf{h}}$* and the *width ${\mathrm{width}}({\mathbf{h}})$ of ${\mathbf{h}}$* by $${\ensuremath{\mathrm{in}}\hspace{1pt}}({\mathbf{h}})=i_{\frac a 2}$$ and $${\mathrm{width}}({\mathbf{h}})=\min \{ i_k+i_{a-k+1}: k=1,2,\dots, \mbox{${a \over 2}$}\}.$$ The following fact is straightforward.

\[dist\] With the same notation as above, one has $${\mathrm{width}}({\mathbf{h}})=
\max\{\ell : h_0+ h_1 +\cdots +h_k \leq h_d+h_{d-1} + \cdots +h_{\ell -k}
\mbox{ for all }k\}.$$

We need the following reformulation of Lemma \[3.A\].

\[kolinsrev\] Let ${\mathbf{h}}=(h_0,h_1,\dots,h_d) \in { \ensuremath{\mathbb{Z}}}_{\geq 0}^{d+1}$ be such that $h_0=0$, $\partial h_i \geq 0$ for $i=0,1,\dots,d-1$ and $\sum_{k=0}^d h_k$ is even. Let $P$ be a constructible $(d-1)$-ball such that $\partial P$ contains $\Delta_d(\ell)$ with $\ell \geq {\ensuremath{\mathrm{in}}\hspace{1pt}}({\mathbf{h}})$. Then there is a constructible $(d-1)$-ball $Q$ such that $h(Q)=h(P)+{\mathbf{h}}$ and $\partial Q$ contains $\Delta_d(\min\{\ell,{\mathrm{width}}({\mathbf{h}})\})$.

We write $${\mathbf{h}}= ({\mathbf{e}}_{i_1} + {\mathbf{e}}_{i_a}) + ({\mathbf{e}}_{i_2} + {\mathbf{e}}_{i_{a-1}}) + \cdots + ({\mathbf{e}}_{i_{\frac a 2}}+{\mathbf{e}}_{i_{\frac a 2 +1}}),$$ where $a=\sum_{k=0}^d h_k$ and where $i_1 \leq i_2 \leq \cdots \leq i_a$. For $k=1,2,\dots,\frac a 2$, let $$w_k=\min \{\ell,i_1+i_a,i_2+i_{a-1},\dots,i_k + i_{a-k+1} \}$$ and let $w_0=\ell$. The assumption $\partial h_i \geq 0$ for $i=0,1,\dots,d-1$ implies that $$\begin{aligned}
\label{XYZ}
i_l+i_{a-l+1} \leq d\end{aligned}$$ for all $l=1,2,\dots, \frac a 2$.

We inductively prove that, for $k=0,1,\dots,\frac a 2$, there is a constructible $(d-1)$-ball $Q_k$ which satisfies the following two conditions:

-   $h(Q_k)=h(P)+({\mathbf{e}}_{i_1} + {\mathbf{e}}_{i_a}) + ({\mathbf{e}}_{i_2} + {\mathbf{e}}_{i_{a-1}}) + \cdots + ({\mathbf{e}}_{i_k}+{\mathbf{e}}_{i_{a-k+1}})$;

-   $\partial Q_k$ contains $\Delta_d(w_k)$.

Since $w_{\frac a 2}=\min\{\ell,{\mathrm{width}}({\mathbf{h}})\}$, if the above statement holds then $Q_{\frac a 2}$ is the simplicial poset with the desired properties.

For $k=0$, $Q_0=P$ satisfies the desired conditions. Suppose that such $Q_k$, where $0\leq k < \frac a 2$, exists. We prove the existence of $Q_{k+1}$. Observe $$i_{k+1} \leq \min\{\ell, i_{a-k}\} \leq w_k,$$ where the last inequality follows from $i_{a-k}\leq i_{a-k+1} \leq \cdots \leq i_a$. Recall that $i_{a-k} \leq d-i_{k+1}$ by and that $\partial Q_k$ contains $\Delta_d(w_k)$ by the induction hypothesis. By Lemma \[3.A\] (apply the case when $P=Q_k$, $i=i_{k+1}$, $m=w_k$ and $j=i_{a-k}$) there is a constructible $(d-1)$-ball $Q_{k+1}$ such that $h(Q_{k+1})=h(Q_k)+ {\mathbf{e}}_{i_{k+1}} + {\mathbf{e}}_{i_{a-k}}$ and $\partial Q_{k+1}$ contains $\Delta_d(\min\{w_k,i_{k+1}+i_{a-k}\})=\Delta_d(w_{k+1})$ as desired.

By considering the special case when $P$ is a Boolean algebra, we obtain the next corollary.

\[kolins2\] Let ${\mathbf{h}}=(h_0,h_1,\dots,h_d) \in { \ensuremath{\mathbb{Z}}}_{\geq 0}^{d+1}$ be such that $h_0=0$, $\partial h_i \geq 0$ for $i=0,1,\dots,d-1$ and $\sum_{k=0}^d h_k$ is even. Then there is a constructible $(d-1)$-ball $P$ such that $h(P)={\mathbf{e}}_0+{\mathbf{h}}$ and $\partial P$ contains $\Delta_d({\mathrm{width}}({\mathbf{h}}))$.

The next lemma is our second main lemma.

\[3.B\] Let $n,m$ and $d$ be positive integers such that $n \leq \frac d 2$ and $d-n \leq m <d$. Let ${\mathbf{h}}=(h_0,h_1,\dots,h_d) \in { \ensuremath{\mathbb{Z}}}^{d+1}$ be a vector such that $\sum_{k=0}^d h_k=d$, $h_i=1$ for $0 \leq i <d-n$, $h_i>0$ for $d-n \leq i \leq m$ and $h_i=0$ for $i>m$. There is a constructible $(d-1)$-ball $P$ such that

-   $h(P)={\mathbf{h}}$, and

-   $\partial P$ contains two ideals $I_1$ and $I_2$ that have no common facets such that $I_1 \cong \Delta_d(n)$ and $I_2 \cong \Delta_d(d-n)$.

Consider the simplicial posets $$A = \big\{{\mathcal{A}}(F): F \subset [d+1],\ F \not \supset [n] \big\} \cong \Delta_{d+1}(n)$$ and $$B = \big\{{\mathcal{B}}(F): F \subset [d+1],\ F \not \supset \{n+1,\dots,d\} \big\} \cong \Delta_{d+1}(d-n),$$ where the ordering is defined by inclusion on $F$. Consider the simplicial complex $$\begin{aligned}
\Sigma\!\! &=&\!\! \big\langle [d+1]\setminus \{i,j\}: i\in [n],\ j\in \{n+1,\dots,d\} \big\rangle\\
&=&\!\! \big\{ F\! \subset\! [d+1]:\! \mbox{ $F \subset [d+1]\setminus \{i,j\}$ for some } i\! \in\! [n] \mbox{ and } j\! \in\! \{n\!+\!1,\dots,d\} \big\}.\end{aligned}$$ Let $$I_1=\big\langle {\mathcal{A}}\big([d+1]\setminus \{i,d+1\}\big): i=1,2,\dots,n\big\rangle \cong \Delta_d(n)$$ and $$I_2=\big\langle {\mathcal{B}}\big([d+1]\setminus \{i,d+1\}\big): i=n+1,n+2,\dots,d \big\rangle \cong \Delta_d(d-n).$$ Then $$\begin{aligned}
\partial A
&=&
\big\langle {\mathcal{A}}\big([d+1]\setminus\{i,j\}\big): i=1,2,\dots,n,\ j=n+1,\dots,d+1\big\rangle\\
&=&\big\{{\mathcal{A}}(F): F \in \Sigma\big\} \cup I_1,\end{aligned}$$ and similarly $$\partial B=\big\{{\mathcal{B}}(F): F \in \Sigma\big\} \cup I_2.$$

Let ${\boldsymbol g}=(g_0,g_1,\dots,g_d)= {\mathbf{h}}-\sum_{i=0}^{n-1} {\mathbf{e}}_i - \sum_{i=0}^{d-n-1} {\mathbf{e}}_i$. Observe that $A$ and $B$ are constructible $(d-1)$-balls with $h(A)=\sum_{i=0}^{n-1} {\mathbf{e}}_i$ and $h(B)=\sum_{i=0}^{d-n-1} {\mathbf{e}}_i$. By Lemma \[3.1\], if there is a constructible $(d-2)$-ball $\Omega \subset \Sigma$ such that $-h_i(\Omega)+h_{i-1}(\Omega)=g_i$ for all $i$, then the simplicial poset $$P=A \cup_\Omega B = (A \cup B) /\! \sim,$$ where $\sim$ is the equivalence relation defined by ${\mathcal{A}}(F) \sim {\mathcal{B}}(F)$ for $F \in \Omega$, satisfies the desired conditions (i) and (ii). Thus the next lemma completes the proof.

\[3.5\] With the same notation as in the proof of Lemma \[3.B\], there is a shellable $(d-2)$-ball $\Omega \subset \Sigma$ such that $-h_i(\Omega)+h_{i-1}(\Omega)=g_i$ for all $i$.

Observe $${\boldsymbol g}=(-1,\dots,-1,0,\dots,0,h_{d-n},h_{d-n+1},\dots,h_d),$$ where $-1$ appears in the first $n$ entries. Let $$\alpha_\ell=h_{(d-n)+\ell-1} + h_{(d-n)+\ell} +\cdots + h_d$$ for $\ell=1,2,\dots,n+1$. By the assumption on ${\mathbf{h}}=(h_0,h_1,\dots,h_d)$, $\alpha_1=n$ and $\alpha_\ell \leq n -(\ell-1) $ for all $\ell$.

Let $$\begin{aligned}
D&=&\big\{\{p,q\}: p\in [n],\ q \in \{n+1,\dots,d\},\ p+q \leq d \big\}\\
&&\bigcup \left[\bigcup_{\ell=1}^{m-(d-n)+1} \big\{ \{p,d+\ell-p\}: p=\ell,\ell+1,\dots,\ell+\alpha_{\ell}-1 \big\} \right].\end{aligned}$$ Since $\alpha_\ell \leq n-(\ell-1)$, $D \subset \{\{p,q\}: p \in [n],\ q\in\{n+1,\dots,d\} \}$. Let $$\Omega=\big\langle [d+1] \setminus \{p,q\}: \{p,q\} \in D \big\rangle \subset \Sigma.$$ We claim that $\Omega$ satisfies the desired conditions.

We first prove that $\Omega$ is shellable. We define the total order $\succ$ on $D$ by $\{p,q\} \succ \{p',q'\}$, where $p<q$ and $p'<q'$, if $p<p'$ or $p=p'$ and $q<q'$. We show that the total order on the facets of $\Omega$ induced by $\succ$ gives a shelling of $\Omega$. Since $h_{n-d},\dots,h_m>0$ and $h_{m+1}= \cdots =h_d=0$ by the assumption, we have $\alpha_1> \alpha_2> \cdots > \alpha _{m-(d-n)+2}= \cdots = \alpha _{n+1}=0$. Then, by the construction of $D$, for any $\{p,q\} \in D$ with $p<q$, we have $\{p-1,q\} \in D$ if $p \ne 1$ and $\{p,q-1\} \in D$ if $q \ne n+1$. This fact shows that, for any $\{p,q\} \in D$ with $\{p,q\} \ne \{1,n+1\}$, $$\begin{aligned}
&&\big\langle [d+1]\setminus \{s,t\}: \{s,t\} \in D,\ \{s,t\} \succ \{p,q\} \big\rangle
\bigcap \big\langle [d+1]\setminus \{p,q\} \big\rangle\\
&&= \big\langle [d+1] \setminus \{p,q,k\}:
k=1,2,\dots,p-1,n+1,n+2,\dots,q-1 \big \rangle\\
&& \cong \Delta_{d-1}(p+q-n-2).\end{aligned}$$ Hence $\Omega$ is shellable. (To see that the second line contains the first line, use the fact that if $F$ is contained in the first line then one has either $p'=\min([n] \setminus F) <p$ or $q'=\min(\{n+1,\dots,d\}\setminus F) <q$ since $[d+1]\setminus \{p',q'\}$ must be contained in $\{ [d+1]\setminus \{s,t\}: \{s,t\} \in D,\ \{s,t\} \succ \{p,q\} \}$.)

The above shelling and Lemma \[shellablehvector\] show $$h_i(\Omega)=\# \big\{\{p,q\} \in D: p+q-n-2=i \big\}$$ for $i=0,1,\dots,d-1$. Then a simple counting shows $$\begin{aligned}
h_i(\Omega)=
\left\{
\begin{array}{ll}
i+1, & \mbox{ if } i \leq n-1,\\
n, & \mbox{ if } n \leq i \leq d-n-1,\\
\alpha_{i-(d-n)+2}, & \mbox{ if }  i \geq d-n.
\end{array}
\right.\end{aligned}$$ Note that $h_{d-n-1}(\Omega)=\alpha_1=n$. Then we have $-h_i(\Omega)+h_{i-1}(\Omega)=g_i$ for all $i$.

Finally, $\Omega$ is a simplicial cell $(d-2)$-ball since $\Omega$ can be identified with a $(d-2)$-dimensional subposet of a $(d-2)$-dimensional simplicial cell sphere $\partial A$.

The simplicial complex $\Sigma$ is the cone of the join of the boundaries of simplexes, and $h$-vectors of shellable subcomplexes of $\Sigma$ having the same dimension as $\Sigma$ are characterized in [@BFS]. This will give an alternative proof of Lemma \[3.5\].

\[sufficiency\] If ${\mathbf{h}}=(h_0,h_1,\dots,h_d) \in { \ensuremath{\mathbb{Z}}}^{d+1}$ satisfies conditions (1)–(7) in Theorem \[1.2\], then ${\mathbf{h}}$ is the $h$-vector of a simplicial cell $(d-1)$-ball.

If $\partial {\mathbf{h}}$ is positive or $\sum_{k=0}^d h_k$ is even, then the assertion was proved in [@Ko Theorem 16]. But we include a proof for completeness (our proof is essentially the same as that of Kolins).

[*Case 1.*]{} Suppose that all entries of $\partial {\mathbf{h}}$ are positive. If $\sum_{k=0}^d h_k$ is odd then, since $\partial({\mathbf{h}}-{\mathbf{e}}_0)$ is a non-negative vector, Corollary \[kolins2\] says that ${\mathbf{h}}={\mathbf{e}}_0+({\mathbf{h}}-{\mathbf{e}}_0)$ is the ${\mathbf{h}}$-vector of a simplicial cell $(d-1)$-ball. Suppose that $\sum_{k=0}^d h_k$ is even. We write $${\mathbf{h}}-{\mathbf{e}}_0=({\mathbf{e}}_{i_1}+{\mathbf{e}}_{i_a})+ ({\mathbf{e}}_{i_2}+{\mathbf{e}}_{i_{a-1}})+ \cdots + ({\mathbf{e}}_{i_{\frac {a-1} 2}}+{\mathbf{e}}_{i_{\frac {a+3} 2 }}) + {\mathbf{e}}_{i_{\frac {a+1} 2 }},$$ where $a=\sum_{k=0}^d h_k -1$ and where $i_1 \leq \cdots \leq i_a$. Let ${\mathbf{h}}'={\mathbf{h}}-{\mathbf{e}}_0 -{\mathbf{e}}_{i_{\frac {a+1} 2 }}$. Since $\partial({\mathbf{h}}-{\mathbf{e}}_0)$ is non-negative, we have $i_{a-l+1} \leq d-i_l$ for $l=1,2,\dots,\frac {a-1} 2$. Then $\partial {\mathbf{h}}'$ is non-negative and ${\mathrm{width}}({\mathbf{h}}') \geq \min\{i_{\frac {a+3} 2 },\dots,i_a\} \geq i_{\frac {a+1} 2}$. Thus, by Corollary \[kolins2\], there is a constructible $(d-1)$-ball $P$ such that $h(P)={\mathbf{e}}_0+{\mathbf{h}}'= {\mathbf{h}}-{\mathbf{e}}_{i_{\frac {a+1} 2}}$ and $\partial P$ contains $\Delta_d(i_{\frac {a+1} 2})$. Let $Q$ be a simplicial cell $(d-1)$-ball obtained from $P$ by gluing a Boolean algebra of rank $d$ along an order ideal $J \subset \partial P$ with $J \cong \Delta_d(i_{\frac {a+1} 2})$. Lemma \[3.2\] guarantees that the $h$-vector of $Q$ is $h(P)+{\mathbf{e}}_{i_{\frac {a+1} 2}}={\mathbf{h}}$.

[*Case 2.*]{} Suppose that $\partial {\mathbf{h}}$ has a zero entry and $\sum_{k=0}^d h_k$ is even. Let $$n=\min\{k :\partial h_k=0\}$$ and let ${\mathbf{h}}'=(h_0',h_1',\dots,h_d')={\mathbf{h}}-{\mathbf{e}}_0-{\mathbf{e}}_{d-n}$. Note that ${\mathbf{h}}'$ is a non-negative vector since $\partial h_{n-1}>0$ and $\partial h_n=0$ imply $h_{d-n} > 0$. By the same argument as in Case 1, it is enough to prove that there is a constructible $(d-1)$-ball $P$ such that $h(P)={\mathbf{h}}-{\mathbf{e}}_{d-n}$ and $\partial P$ contains $\Delta_d(d-n)$. By the choice of $n$, $\partial {\mathbf{h}}'$ is non-negative. Thus, by Corollary \[kolins2\], what we must prove is ${\mathrm{width}}({\mathbf{h}}') \geq d-n$.

By the condition (4), for $k \geq 0$, we have $$h_{k+n}+ \cdots + h_{k+1}
\geq
(h_0+ h_1+ \cdots + h_{k+n}) - (h_d+ h_{d-1}+ \cdots + h_{d-k-n})$$ (we substitute $k$ by $k+n$ to the inequality in (4)). Then we have $h_0+\cdots+h_{k} \leq h_d+ \cdots + h_{d-k-n}$ for all $k\geq 0$. This fact shows $$h_0'+h_1'+\cdots+h_{k}' 
\leq
h_d'+ h_{d-1}'+ \cdots + h'_{d-k-n}$$ for all $k \geq 0$. Then Lemma \[dist\] guarantees ${\mathrm{width}}({\mathbf{h}}') \geq d-n$.

[*Case 3.*]{} Suppose that $\partial {\mathbf{h}}$ has a zero entry and $\sum_{k=0}^d h_k$ is odd. By the condition (3), $d$ must be even. Let $$n=\min\{k :\partial h_k=0\}.$$ By the symmetry of $\partial {\mathbf{h}}$, we have $n < \frac d 2$. Let $$m=\min\{k: h_k =0\}-1.$$ Thus $h_0,h_1,\dots,h_m>0$ and $h_{m+1}=0$. Then $m \leq d-1$ since $h_d=0$ and $m \geq d-n$ by the condition (5).

By the condition (7), $\partial h_{d-m-1} \geq d-m$. Recall that $\partial h_n=0$, $d-m \leq n$ and $\partial h_k > 0$ for $k<n$. Then, there is a sequence of integers $d-m \leq s_1 <s_2 < \cdots < s_p =n$ such that

-   $d-m > \partial h_{s_1} > \cdots > \partial h_{s_p}=0$, and

-   for any $s_{i-1} <j < s_i$ with $i \geq 2$ one has $\partial h_j \geq \partial h_{s_{i-1}}$, and for any $d-m -1 <j < s_1$ one has $\partial h_j \geq d-m$.

(The next figure explains what $s_1,s_2,\dots,s_p$ are.)

0.1in

( 50.1300, 17.5600)( 7.9800,-25.5600)

(14.2800,-10.5100)[(0,0)[$d\!-\!m$]{}]{}(16.2800,-6.9000)[(0,0)[$\partial h_i$]{}]{}

(14.2800,-13.0200)[(0,0)[$\partial h_{s_1}$]{}]{}(14.2800,-16.7800)[(0,0)[$\partial h_{s_2}$]{}]{}(14.2800,-20.5400)[(0,0)[$\partial h_{s_3}$]{}]{}(22.6400,-24.3000)[(0,0)[$d\!-\!m\!-\!1$]{}]{}(28.9200,-24.3000)[(0,0)[$s_1$]{}]{}(37.2800,-24.3000)[(0,0)[$s_2$]{}]{}(45.6400,-24.3000)[(0,0)[$s_3$]{}]{}(51.9100,-24.3000)[(0,0)[$s_p$]{}]{}(55.0100,-23.0000)[(0,0)[$i$]{}]{}(18.2000,-24.3200)[(0,0)[$\cdots$]{}]{}

Let $${\boldsymbol \gamma}=(d-m-\partial h_{s_1}) {\mathbf{e}}_{d-s_1} + \sum_{j=2}^p (\partial h_{s_{j-1}}- \partial h_{s_j} ){\mathbf{e}}_{d-s_j}$$ and $${\boldsymbol \delta}=(\delta_0,\delta_1,\dots,\delta_d) = \sum_{i=0}^m {\mathbf{e}}_i + {\boldsymbol \gamma}.$$ By the construction of ${\boldsymbol \delta}$, $\sum_{k=0}^d \delta_k=d+1$ and $\delta_{d-n} \geq 2$. Let $$\overline {\boldsymbol \delta} = {\boldsymbol \delta} -{\mathbf{e}}_{d-n}.$$ Then $\overline {\boldsymbol \delta}$ satisfies the assumption of Lemma \[3.B\]. Thus there is a constructible $(d-1)$-ball $P$ such that

-   $h(P)=\overline {\boldsymbol \delta}$.

-   $\partial P$ contains two ideals $I_1$ and $I_2$ that have no common facets such that $I_1 \cong \Delta_d(n)$ and $I_2 \cong \Delta_d(d-n)$.

We define ${\mathbf{h}}'=(h'_0,h_1',\dots,h'_d) \in { \ensuremath{\mathbb{Z}}}^{d+1}$ and ${\mathbf{h}}''=(h''_0,h_1'',\dots,h''_d) \in { \ensuremath{\mathbb{Z}}}^{d+1}$ by $${\mathbf{h}}'= (0,\dots,0,h_{n+1}-1,\dots,h_{d-n-1}-1,0,\dots,0)$$ and $${\mathbf{h}}''=  (h_0-1,\dots,h_n-1,0,\dots,0,h_{d-n}-1,\dots,h_m-1,h_{m+1},\dots,h_d)
- {\boldsymbol \gamma}.$$ We claim that the following conditions hold.

-   ${\mathbf{h}}={\mathbf{h}}'+{\mathbf{h}}''+ \overline {\boldsymbol \delta} + {\mathbf{e}}_{d-n}$;

-   ${\mathbf{h}}'$ and ${\mathbf{h}}''$ are non-negative;

-   $\sum_{k=0}^d h_k'$ and $\sum_{k=0}^d h''_k$ are even;

-   $\partial {\mathbf{h}}'$ and $\partial {\mathbf{h}}''$ are non-negative and $\partial h''_n=0$;

-   ${\mathrm{width}}({\mathbf{h}}') \geq d-n$.

We first prove that the above conditions (i)–(v) prove the desired statement.

By Corollary \[kolins2\] and conditions (ii)–(v), there is a constructible $(d-1)$-ball $Q$ with $h(Q)={\mathbf{e}}_0+{\mathbf{h}}'$ such that $\partial Q$ contains an order ideal $J$ which is isomorphic to $\Delta_d(d-n)$. Then the simplicial poset $$R=Q {{}_J \cup_{I_2}} P$$ is a constructible $(d-1)$-ball with $h(R)={\mathbf{h}}'+ \overline {\boldsymbol \delta} +{\mathbf{e}}_{d-n}$ by Lemma \[3.2\]. Also, $\partial R$ contains $\Delta_d(n)$ since $I_1$ and $I_2$ have no common facets.

Since ${\mathbf{h}}''$ is non-negative and $\sum_{k=0}^d h_k''$ is even, ${\mathbf{h}}''$ can be written in the form $${\mathbf{h}}''=({\mathbf{e}}_{i_1}+ {\mathbf{e}}_{i_a}) +({\mathbf{e}}_{i_2}+ {\mathbf{e}}_{i_{a-1}}) + \cdots +({\mathbf{e}}_{i_{\frac a 2}}+ {\mathbf{e}}_{i_{\frac a 2+1}})$$ where $a=\sum_{k=0}^d h_k''$ and $i_1 \leq \cdots \leq i_a$. Since $\partial h''_n=0$, we have $$i_1 \leq \cdots \leq i_{\frac a 2} \leq n \mbox{ and } d-n \leq i_{\frac a 2 +1} \leq \cdots \leq i_a.$$ Thus ${\ensuremath{\mathrm{in}}\hspace{1pt}}({\mathbf{h}}'') \leq n$. Since $\partial R$ contains $\Delta_d(n)$, Lemma \[kolinsrev\] guarantees the existence of a constructible $(d-1)$-ball whose $h$-vector is $h(R)+{\mathbf{h}}''={\mathbf{h}}$.

It remains to prove (i)–(v). Statement (i) is obvious.

[*Proof of (ii).*]{} Since $h_k > 0$ for $k \leq d-n$ (see the first paragraph of Case 3), ${\mathbf{h}}'$ is non-negative. To prove that ${\mathbf{h}}''$ is non-negative, it is enough to prove that $\delta_k \leq h_k$ for all $k$. For $k > m$, $h_k \geq 0=\delta_k$. Also, if $k \leq m$ and $k \not \in \{d-s_1,\dots,d-s_p\}$ then $h_k \geq 1 = \delta_k$ by the choice of $m$. Suppose $k=d-s_j$. Since $h_{s_j} - h_{d-s_j}= \partial h_{s_j} - \partial h_{s_j-1}$, $$h_{d-s_j} = h_{s_j} +(\partial h_{s_j-1}-\partial h_{s_j}) 
\geq 1+(\partial h_{s_{j-1}} -\partial h_{s_j}) = \delta_{d-s_j},$$ where the inequality follows from the choice of $s_1,\dots,s_p$. (If $j=1$ then we consider that $\partial h_{s_0}=d-m$.)

[*Proof of (iii).*]{} Since $\partial h_n=(h_0+\cdots+h_n) - (h_d + \cdots +h_{d-n})=0$ and since $h_0+h_1+ \cdots +h_d$ is odd, $$h_{n+1} + \cdots + h_{d-n-1}= (h_0+ \cdots +h_d) -(h_0+ \cdots+h_n)-(h_d+ \cdots + h_{d-n})$$ is odd. Then, since $d$ is even, $$(h_0'+ \cdots + h_d') = (h_{n+1} + \cdots + h_{d-n-1}) -(d-2n-1)$$ is even, as desired. Similarly, $\partial h_n=0$ implies that $$(h_0''+ \cdots + h_d'')=
(h_0+\cdots+h_n)+(h_d+ \cdots +h_{d-n})-2n-2$$ is even.

[*Proof of (iv).*]{} Observe $\partial h'_k= \partial h_k - \partial h_n$ for $k \geq n$. Since $\partial h_n=0$, we have $$\begin{aligned}
\partial h_k'=
\left\{
\begin{array}{lll}
0, & \mbox{ for } k=0,1,\dots,n,\\
\partial h_k, & \mbox{ for } k=n+1,\dots,\frac d 2 -1.\\
\end{array}
\right.\end{aligned}$$ Hence $\partial {\mathbf{h}}'$ is non-negative.

For $k=0,1,\dots,d-m-1$, $$\partial h''_k =\partial h_k -(k+1) \geq 0$$ by the condition (7). Also, for $k=n,\dots,\frac d 2 -1$, $$\partial h''_k=\partial h_n=0.$$ Let $k \in \{d-m,d-m+1,\dots,n-1\}$. Then, there is an $i$ such that $s_{i-1} \leq k < s_i$, where we consider that $s_0=d-m$ and $\partial h_{s_0}=d-m$. Then $$\begin{aligned}
\partial h''_k&=&
\partial h_k -(d\!-\!m) + \{ (d\!-\!m\!-\! \partial h_{s_1}) + (\partial h_{s_1}\! - \!\partial h_{s_2}) + \cdots + (\partial h_{s_{i-2}}\!-\!\partial h_{s_{i-1}})\} \\
& = & \partial h_k - \partial h_{s_{i-1}}\\
&\geq& 0,\end{aligned}$$ where the last inequality follows from the choice of $s_1,\dots,s_p$. Thus $\partial {\mathbf{h}}''$ is non-negative.

[*Proof of (v).*]{} The condition (6) says that, for $0  \leq  k \leq d-2n$, $$\begin{aligned}
h_{k+n}+ \cdots + h_{k+1} -n 
&\geq& \partial h_{k+n}\\
&=& \partial h_n + (h_{n+1}+ \cdots + h_{n+k}) -(h_{d-n-1}+ \cdots + h_{d-n-k})\end{aligned}$$ (we substitute $\ell$ by $k+n$ in the inequality in (6)). Since $\partial h_n=0$, for $n<k  \leq d-2n$, we have $$h_{n+1} + \cdots + h_{k} +n\leq h_{d-n-1} + \cdots + h_{d-n-k}.$$ The above inequality says $$\begin{aligned}
h_0'+ \cdots + h_{k}' &=&h_{n+1} + \cdots + h_{k} -(k-n)\\
& \leq& h_{d-n-1} + \cdots + h_{d-n-k} -k
 = h_d' + \cdots + h_{d-n-k}'\end{aligned}$$ for all $n<k<d-2n$. Also, if $k \leq n$ or $k \geq d-2n$ then it is clear that $$h_0'+ \cdots + h_{k}'\leq  h_d' + \cdots + h_{d-n-k}'.$$ Then Lemma \[dist\] guarantees ${\mathrm{width}}({\mathbf{h}}')\geq d-n$.

**Acknowledgments**: I would like to thank Samuel Kolins for his valuable comments on my earlier work, which inspired me to find the complete necessary and sufficient conditions, and for pointing out a mistake in an earlier draft. This work was supported by KAKENHI 22740018.

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