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A008300
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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.
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24
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1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 24, 90, 24, 1, 1, 120, 2040, 2040, 120, 1, 1, 720, 67950, 297200, 67950, 720, 1, 1, 5040, 3110940, 68938800, 68938800, 3110940, 5040, 1, 1, 40320, 187530840, 24046189440, 116963796250, 24046189440, 187530840, 40320, 1, 1, 362880, 14398171200, 12025780892160, 315031400802720, 315031400802720, 12025780892160, 14398171200, 362880, 1
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OFFSET
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0,5
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COMMENTS
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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.
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 236, P(n,k).
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LINKS
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Brendan D. McKay, Rows n = 0..30, flattened
C. J. Everett and P. R. Stein, The asymptotic number of integer stochastic matrices, Disc. Math. 1 (1971), 55-72.
Richard J. Mathar, 2-regular Digraphs of the Lovelock Lagrangian, arXiv:1903.12477 [math.GM], 2019.
B. D. McKay, Applications of a technique for labeled enumeration, Congress. Numerantium, 40 (1983), 207-221.
Brendan D. McKay, first 30 rows : entries named Bv[n,k,n,k]
Wouter Meeussen, relevant entries from B. D. McKay reference
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FORMULA
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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.
T(n,k) = T(n,n-k).
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EXAMPLE
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Triangle begins:
1;
1, 1;
1, 2, 1;
1, 6, 6, 1;
1, 24, 90, 24, 1;
1, 120, 2040, 2040, 120, 1;
1, 720, 67950, 297200, 67950, 720, 1;
1, 5040, 3110940, 68938800, 68938800, 3110940, 5040, 1;
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PROG
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(PARI)
T(n, k)={
local(M=Map(Mat([n, 1])));
my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
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))));
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]);
} \\ Andrew Howroyd, Apr 03 2020
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CROSSREFS
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Row sums give A067209.
Central coefficients are A058527.
Cf. A000142 (column 1), A001499 (column 2), A001501 (column 3), A058528 (column 4), A075754 (column 5), A172544 (column 6), A172541 (column 7).
Cf. A133687, A333157.
Sequence in context: A322620 A155795 A009963 * A321789 A173887 A288025
Adjacent sequences: A008297 A008298 A008299 * A008301 A008302 A008303
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KEYWORD
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tabl,nonn,nice
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Greg Kuperberg, Feb 08 2001
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STATUS
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approved
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