Constant row-column sum matrices? 
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*Given an integer $T$ how many $n\times n$ matrices in $\mathbb Z_{\geq0}^{n\times n}$ we have such that every row and column sum to same $T$?

*Do the set of constant row and column sum matrices form any kind of algebraic structure (it is definitely definable by linear equations) by which I mean closedness, more than permutation invariance at least for special classes etc.)?
 A: If you are looking for a simple formula, you are out of luck except for small $n$ or $T$. As Sam mentions in a comment, these are the integer points in a dilated Birkhoff polytope. Since the vertices of the polytope are lattice points, the number of matrices is a polynomial of degree $(n-1)^2$ in $T$ for any fixed $n$ (the Ehrhart polynomial) but the polynomial is only known up to $n=9$.
This paper and this paper have some stuff about asymptotics.  Searching for these buzzwords will get you more interesting things, like this paper.
A: I commented there's closure under (matrix) multiplication, and OP asked for an explanation, so here goes. 
Let $A,B$ be $n\times n$ constant line-sum matrices with constants $a,b$, respectively. 
Let $A$ have rows $r_1,r_2,\dots,r_n$, let $B$ have columns $c_1,c_2,\dots,c_n$, let $C=AB$, and let $\underline b$ be the column $n$-vector $$\pmatrix{b\cr b\cr\vdots\cr b\cr}$$
Then the entries of the first row of $C$ are the dot products $r_1\cdot c_1,r_1\cdot c_2,\dots,r_1\cdot c_n$. So, the sum of the entries of the first row of $C$ is $$r_1\cdot(c_1+c_2+\cdots+c_n)=r_1\cdot\underline{b}=ab$$ 
The same argument applies to each row of $C$, and a similar argument applies to each column of $C$, so $C$ is a constant line-sum matrix with constant $ab$. 
