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A008300 Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= n) gives number of {0,1} n X n matrices with all row and column sums equal to k.

%I A008300 #65 Jul 04 2023 08:41:42
%S A008300 1,1,1,1,2,1,1,6,6,1,1,24,90,24,1,1,120,2040,2040,120,1,1,720,67950,
%T A008300 297200,67950,720,1,1,5040,3110940,68938800,68938800,3110940,5040,1,1,
%U A008300 40320,187530840,24046189440,116963796250,24046189440,187530840,40320,1,1,362880,14398171200,12025780892160,315031400802720,315031400802720,12025780892160,14398171200,362880,1
%N A008300 Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= n) gives number of {0,1} n X n matrices with all row and column sums equal to k.
%C A008300 Or, triangle of multipermutation numbers T(n,k), n >= 0, 0 <= k <= n: number of relations on an n-set such that all vertical sections and all horizontal sections have k elements.
%D A008300 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 236, P(n,k).
%H A008300 Brendan D. McKay, <a href="/A008300/b008300.txt">Rows n = 0..30, flattened</a>
%H A008300 C. J. Everett and P. R. Stein, <a href="https://dx.doi.org/10.1016/0012-365X(71)90007-0">The asymptotic number of integer stochastic matrices</a>, Disc. Math. 1 (1971), 55-72.
%H A008300 Richard J. Mathar, <a href="https://arxiv.org/abs/1903.12477">2-regular Digraphs of the Lovelock Lagrangian</a>, arXiv:1903.12477 [math.GM], 2019.
%H A008300 Richard J. Mathar, <a href="https://vixra.org/abs/2306.0157">Rencontres for equipartite distributions of multisets of colored balls into urns</a>, vixra:2306.0157 (2023)
%H A008300 B. D. McKay, <a href="http://users.cecs.anu.edu.au/~bdm/papers/LabelledEnumeration.pdf">Applications of a technique for labeled enumeration</a>, Congress. Numerantium, 40 (1983), 207-221.
%H A008300 Brendan D. McKay, <a href="http://users.cecs.anu.edu.au/~bdm/data/Bvals.txt">first 30 rows : entries named Bv[n,k,n,k]</a>
%H A008300 Wouter Meeussen, <a href="/A008300/a008300.txt">relevant entries from B. D. McKay reference</a>
%F A008300 Comtet quotes Everett and Stein as showing that T(n,k) ~ (kn)!(k!)^(-2n) exp( -(k-1)^2/2 ) for fixed k as n -> oo.
%F A008300 T(n,k) = T(n,n-k).
%e A008300 Triangle begins:
%e A008300   1;
%e A008300   1,    1;
%e A008300   1,    2,       1;
%e A008300   1,    6,       6,        1;
%e A008300   1,   24,      90,       24,        1;
%e A008300   1,  120,    2040,     2040,      120,       1;
%e A008300   1,  720,   67950,   297200,    67950,     720,    1;
%e A008300   1, 5040, 3110940, 68938800, 68938800, 3110940, 5040, 1;
%e A008300   ...
%o A008300 (PARI)
%o A008300 T(n, k)={
%o A008300   local(M=Map(Mat([n, 1])));
%o A008300   my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
%o A008300   my(recurse(i, p, v, e) = if(i<0, if(!e, acc(p, v)), my(t=polcoef(p,i)); for(j=0, min(t, e), self()(i-1, p+j*(x-1)*x^i, binomial(t, j)*v, e-j))));
%o A008300   for(r=1, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(k-1, src[i, 1], src[i, 2], k))); vecsum(Mat(M)[,2]);
%o A008300 } \\ _Andrew Howroyd_, Apr 03 2020
%Y A008300 Row sums give A067209.
%Y A008300 Central coefficients are A058527.
%Y A008300 Cf. A000142 (column 1), A001499 (column 2), A001501 (column 3), A058528 (column 4), A075754 (column 5), A172544 (column 6), A172541 (column 7), A172536 (column 8), A172540 (column 9), A172535 (column 11), A172534 (column 12), A172538 (column 13), A172537 (column 14).
%Y A008300 Cf. A133687, A333157 (symmetric matrices), A257493 (nonnegative elements), A260340 (up to row permutations), A364068 (traceless).
%K A008300 tabl,nonn,nice
%O A008300 0,5
%A A008300 _N. J. A. Sloane_
%E A008300 More terms from _Greg Kuperberg_, Feb 08 2001