interesting matrix Let be $a(k,m),k,m\geq 0$ an infinite matrix then the set
     $$T_k=\{(a(k,0),a(k,1),...,a(k,i),...),(a(k,0),a(k+1,1),...,a(k+i,i),...)\}$$is called angle of matrix
$a(k,0)$ is edge of $T_k$
$a(k,i),a(k+i,i),i>0$ are conjugate elements of $T_k$
$(a(k,0),a(k,1),...,a(k,i),...)$ is horizontal ray of $T_k$
$(a(k,0),a(k+1,1),...,a(k+i,i),...)$is diagonal ray of $T_k$ 
Elements of diagonal ray of $T_0$   are $1$
Elements above diagonal ray of $T_0$   are  $0$ 
Elements of edge of $T_k,k>0$ are $0$
Each element of diagonal ray of $T_k,k>0$ is sum of his conjugate and elements of horizontal ray of $T_k$ that are placed on left.
Prove that sum of elements of row $k$ is partition function $p(k)$  

This is a very unnecessarily complicated, ambiguous and partly erroneous reformulation of a simple recurrence relation for the number $a(k,m)$ of partitions of $k$ with greatest part $m$. It works out if the following changes and interpretations are made:


*

*both instances of $k\gt1$ are replaced by $k\ge1$,

*"his conjugate" is interpreted as "its upper conjugate" (each entry has two conjugates), and

*"on the left" is interpreted as "to the left of its upper conjugate".


The resulting recurrence relation is
$$
a(k,m)=\sum_{i=1}^ma(k-m,i)\;,
$$
which simply states that a partition of $k$ with greatest part $m$ arises by adding a part $m$ to any partition of $k-m$ with greatest part not greater than $m$.
