The number of the partition of the set $A$ into $k$ bounded blocks. Let $A=\{1,2,\cdots,n\}$ be a set. We want to partitions of this set into $k$   non-empty unlabelled subsets $B_1,B_2,\cdots ,B_k$ such that cardinality of each $B_i$ between positive integers $a$ and $b$, that means $a\leq |B_i|\leq b$.
Let $D_{a,b}(n,k)$ be the number of partitions of the set $A$ into $k$   non-empty unlabelled subsets $B_1,B_2,\cdots ,B_k$  which $a\leq |B_i|\leq b$.
How can calculate the number of such partitions?
I try obtained the recurrence relation for $D_{a,b}(n,k)$ with the definition of Stirling numbers of the second kind but I couldn't.
Very thanks for any help and comment.

Supposing that  we are trying  to generalize Stirling numbers  here we
get the combinatorial class
$$\def\textsc#1{\dosc#1\csod}
\def\dosc#1#2\csod{{\rm #1{\small #2}}}
\textsc{SET}_{=k}(\textsc{SET}_{a\le\cdot\le b}(\mathcal{Z}))$$
which yields the generating function
$$G(z) = \frac{1}{k!} \left(\sum_{q=a}^b \frac{z^q}{q!}\right)^k.$$
Differentiate to obtain
$$G'(z) = \frac{1}{(k-1)!} \left(\sum_{q=a}^b \frac{z^q}{q!}\right)^{k-1}
\sum_{p=a-1}^{b-1} \frac{z^p}{p!}.$$
Extracting coefficients we find
$$D_{a,b}(n+1,k) =
n! [z^n] \sum_{p=a-1}^{b-1} \frac{z^p}{p!}
\frac{1}{(k-1)!} \left(\sum_{q=a}^b \frac{z^q}{q!}\right)^{k-1}
\\ = n! \sum_{p=a-1}^{b-1} \frac{1}{p!}
[z^{n-p}] \frac{1}{(k-1)!} 
\left(\sum_{q=a}^b \frac{z^q}{q!}\right)^{k-1}
\\ = n! \sum_{p=a-1}^{b-1} \frac{1}{p!}
\frac{1}{(n-p)!} D_{a,b}(n-p, k-1)
\\ = \sum_{p=a-1}^{b-1} {n\choose p} D_{a,b}(n-p, k-1).$$
The base cases here  are $D_{a,b}(n, k) = 0$ if $n\lt  1$ or $k=0$ and
$D_{a,b}(n,  1) =  [[a\le  n\le b]]$  where we  have  used an  Iverson
bracket.
These were verified using  enumeration, coefficient extraction from
$G(z)$, and the recurrence.  We  also checked that $D_{1,n}(n,k)$ does
indeed produce  Stirling numbers.  (Use the standard  recurrence which
exploits  differentiation  in a  different  way  if you  need  regular
Stirling numbers only.) This was the code.

with(combinat);

ENUM :=
proc(n, k, a, b)
local res, part, mset, inrange, psize;

    res := 0;

    part := firstpart(n);

    while type(part, list) do
        inrange := select(p -> a <= p and p <= b, part);
        psize := nops(part);

        if nops(inrange) = psize and k = psize then
            mset := convert(part, `multiset`);
            res := res +
            n!/mul(p!, p in part)
            /mul(q[2]!, q in mset);
        fi;

        part := nextpart(part);
    od;


    res;
end;

GCF := (n, k, a, b)
-> n!*coeftayl(add(z^q/q!, q=a..b)^k/k!, z=0, n);

DX :=
proc(n, k, a, b)
option remember;

    if n < 1 or k = 0 then return 0 fi;
    if k = 1 then
        if a <= n and n <= b then
            return 1;
        fi;
        return 0;
    fi;

    add(binomial(n-1, p)*DX(n-1-p, k-1, a, b), p=a-1..b-1);
end;

ST2 := (n, k) -> DX(n, k, 1, n);

Remark.  These  data  are  available at  the  OEIS,  consult  e.g.
sequences                         A059022,
A059023,
A059024,                               and
A059025.
