Generalization of permutation matrix For integers $n$ and $k$, I am interested in $n\times n$ matrices with exactly $k$ non-zero entries in each row and each column. The case $k=1$ corresponds to (generalized) permutation matrices. 
In what context have these matrices been studied? Also, is there an easy formula for the total number of such matrices (if the non-zero entries are restricted to be one)? For $k=1$, the answer would be $n!$.
For small values of $n$ and $k=1,\ldots,n$, I find


*

*$n=1$: 1

*$n=2$: 2, 1

*$n=3$: 6, 6, 1

*$n=4$: 24, ...


OEIS finds these sequences; they are related to partitions of $[n]$ into sublists but don't directly mention an obvious connection to my problem.
 A: One other interpretation of these objects are (labelled) $k$-regular bipartite graphs on the vertex set $\{u_1,u_2,\ldots,u_n\} \cup \{v_1,v_2,\ldots,v_n\}$.  The bijection: if cell $(i,j)$ in the $(0,1)$-matrix contains a $1$, we add an edge between $u_i$ and $v_j$.  The $(0,1)$-matrix is known as the biadjacency matrix (or even just adjacency matrix if the context is clear).
They are also a special case of frequency squares and contingency tables.
A: OEIS knows this triangular table as A00830; you should have looked for ´1, 1,2,1, 1,6,6,1, 1,24´ reflecting (pun) the symmetry within each row as starting from k=0.
A: This is an answer to my own question. Apparently, the problem described in the question is open, and has been considered in the more general form of finding the number of $n\times n$ $(0,1)$-matrices with prescribed row and column sums.
Three recent references I found are:


*

*Pérez-Salvador, Blanca Rosa; de-los-Cobos-Silva, Sergio; Gutiérrez-Andrade, Miguel Angel; Torres-Chazaro, Adolfo. A reduced formula for the precise number of $(0,1)$-matrices in ${\scr A}({\bf R},{\bf S})$. Discrete Math. 256 (2002), no. 1-2, 361--372. MR1927558

*Wang, Bo-Ying; Zhang, Fuzhen. On the precise number of $(0,1)$-matrices in ${\scr A}(R,S)$. Discrete Math. 187 (1998), no. 1-3, 211--220. MR1630720

*Tan, Zhonghua; Gao, Shanzhen; Niederhausen, Heinrich. Enumeration of $(0,1)$-matrices with constant row and column sums. Appl. Math. J. Chinese Univ. Ser. B 21 (2006), no. 4, 479--486. MR2270986
Using Mathematica, the number of $n\times n$ $(0,1)$-matrices with row and column sum equal to $k$ can be computed by brute force via
n = 5; k = 2;
Length@Select[
Flatten /@ Flatten[#, n - 1] &@
Outer[List, Sequence @@ ConstantArray[Subsets[Range[n], {k}], n],1],
(Function[x, Count[#, x]] /@ Range[n] == ConstantArray[k, n])&]

