A008300 - OEIS

%I #65 Jul 04 2023 08:41:42

%S 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 297200,67950,720,1,1,5040,3110940,68938800,68938800,3110940,5040,1,1,

%U 40320,187530840,24046189440,116963796250,24046189440,187530840,40320,1,1,362880,14398171200,12025780892160,315031400802720,315031400802720,12025780892160,14398171200,362880,1

%N 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 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 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 236, P(n,k).

%H Brendan D. McKay, <a href="/A008300/b008300.txt">Rows n = 0..30, flattened</a>

%H 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 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 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 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 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 Wouter Meeussen, <a href="/A008300/a008300.txt">relevant entries from B. D. McKay reference</a>

%F 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 T(n,k) = T(n,n-k).

%e Triangle begins:

%e   1;

%e   1,    1;

%e   1,    2,       1;

%e   1,    6,       6,        1;

%e   1,   24,      90,       24,        1;

%e   1,  120,    2040,     2040,      120,       1;

%e   1,  720,   67950,   297200,    67950,     720,    1;

%e   1, 5040, 3110940, 68938800, 68938800, 3110940, 5040, 1;

%e   ...

%o (PARI)

%o T(n, k)={

%o   local(M=Map(Mat([n, 1])));

%o   my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));

%o   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   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 } \\ _Andrew Howroyd_, Apr 03 2020

%Y Row sums give A067209.

%Y Central coefficients are A058527.

%Y 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 Cf. A133687, A333157 (symmetric matrices), A257493 (nonnegative elements), A260340 (up to row permutations), A364068 (traceless).

%K tabl,nonn,nice

%O 0,5

%A _N. J. A. Sloane_

%E More terms from _Greg Kuperberg_, Feb 08 2001