A257493
Number A(n,k) of n X n nonnegative integer matrices with all row and column sums equal to k; square array A(n,k), n >= 0, k >= 0, read by antidiagonals.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 6, 1, 1, 1, 4, 21, 24, 1, 1, 1, 5, 55, 282, 120, 1, 1, 1, 6, 120, 2008, 6210, 720, 1, 1, 1, 7, 231, 10147, 153040, 202410, 5040, 1, 1, 1, 8, 406, 40176, 2224955, 20933840, 9135630, 40320, 1, 1, 1, 9, 666, 132724, 22069251, 1047649905, 4662857360, 545007960, 362880, 1
Offset: 0
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, 7, ...
1, 6, 21, 55, 120, 231, 406, ...
1, 24, 282, 2008, 10147, 40176, 132724, ...
1, 120, 6210, 153040, 2224955, 22069251, 164176640, ...
1, 720, 202410, 20933840, 1047649905, 30767936616, 602351808741, ...
- Alois P. Heinz, Antidiagonals n = 0..20, flattened
- E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce, A generalization of Eulerian numbers via rook placements, arXiv:1508.03673 [math.CO], 2015.
- D. M. Jackson & G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4.4 (1975), 474-477. (Annotated scanned copy)
- Richard J. Mathar, 2-regular Digraphs of the Lovelock Lagrangian, arXiv:1903.12477 [math.GM], 2019.
- Dennis Pixton, Ehrhart polynomials for n = 1, ..., 9
- M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements, Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970. [Annotated scanned copy]
-
with(numtheory):
b:= proc(n, k) option remember; `if`(n=1, 1, add(
`if`(bigomega(d)=k, b(n/d, k), 0), d=divisors(n)))
end:
A:= (n, k)-> b(mul(ithprime(i), i=1..n)^k, k):
seq(seq(A(n, d-n), n=0..d), d=0..8);
-
b[n_, k_] := b[n, k] = If[n==1, 1, Sum[If[PrimeOmega[d]==k, b[n/d, k], 0], {d, Divisors[n]}]]; A[n_, k_] := b[Product[Prime[i], {i, 1, n}]^k, k]; Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 20 2016, after Alois P. Heinz *)
-
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(h, p, q, v, e) = if(!p, if(!e, acc(q, v)), my(i=poldegree(p), t=pollead(p)); self()(k, p-t*x^i, q+t*x^i, v, e); for(m=1, h-i, for(j=1, min(t, e\m), self()(if(j==t, k, i+m-1), p-j*x^i, q+j*x^(i+m), binomial(t, j)*v, e-j*m)))));
for(r=1, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(k, src[i, 1], 0, src[i, 2], k))); vecsum(Mat(M)[, 2])
} \\ Andrew Howroyd, Apr 04 2020
-
bigomega = sloane.A001222
@cached_function
def b(n, k):
if n == 1:
return 1
return sum(b(n//d, k) if bigomega(d) == k else 0 for d in n.divisors())
def A(n, k):
return b(prod(nth_prime(i) for i in (1..n))^k, k)
[A(n, d-n) for d in (0..10) for n in (0..d)] # Freddy Barrera, Dec 27 2018, translated from Maple
-
from sage.combinat.integer_matrices import IntegerMatrices
[IntegerMatrices([d-n]*n, [d-n]*n).cardinality() for d in (0..10) for n in (0..d)] # Freddy Barrera, Dec 27 2018
A001496
Number of 4 X 4 matrices with nonnegative integer entries and row and column sums equal to n.
Original entry on oeis.org
1, 24, 282, 2008, 10147, 40176, 132724, 381424, 981541, 2309384, 5045326, 10356424, 20158151, 37478624, 66952936, 115479776, 193077449, 313981688, 498033282, 772409528, 1173759851, 1750812624, 2567527260, 3706873040
Offset: 0
- L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 124, #25, Q(4,r).
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, pages 233-234.
- M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements. Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970.
- T. D. Noe, Table of n, a(n) for n = 0..1000
- A. G. Bell, Partitioning integers in n dimensions, The Computer Journal, 13 (1970), 278-283.
- Brian Conrey and Alex Gamburd, Pseudomoments of the Riemann zeta-function and pseudomagic squares, Journal of Number Theory, Volume 117, Issue 2, April 2006, Pages 263-278.
- I. J. Good, On the application of symmetric Dirichlet distributions and their mixtures to contingency tables, Ann. Statist. 4 (1976), no. 6, 1159-1189.
- I. J. Good, On the application of symmetric Dirichlet distributions and contingency tables, pp. 1178-1179. (Annotated scanned copy)
- D. M. Jackson and G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4 (1975), 474-477.
- D. M. Jackson & G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4.4 (1975), 474-477. (Annotated scanned copy)
- M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements, Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970. [Annotated scanned copy]
- J. N. van Rijn, F. W. Takes, J. K. Vis, Computing and Predicting Winning Hands in the Trick-Taking Game of Klaverjas, 30th Benelux Conference on Artificial Intelligence (BNAIC 2018), 's-Hertogenbosch, the Netherlands.
- Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
See
A002721 for a 3-dimensional analog.
-
CoefficientList[Series[(1 + 14*x + 87*x^2 + 148*x^3 + 87*x^4 + 14*x^5 + x^6)/(1 - x)^10, {x, 0, 30}], x] (* Wesley Ivan Hurt, Jan 24 2017 *)
LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,24,282,2008,10147,40176,132724,381424,981541,2309384},30] (* Harvey P. Dale, Jul 12 2017 *)
-
x='x+O('x^99); Vec((1+14*x+87*x^2+148*x^3+87*x^4+14*x^5+x^6)/(1-x)^10) \\ Altug Alkan, Apr 17 2016
A058391
Number of 5 X 5 matrices with nonnegative integer entries and all row sums equal to n, up to row and column permutation.
Original entry on oeis.org
1, 7, 198, 5929, 145168, 2459994, 30170387, 282159907, 2114430613, 13190940964, 70598379694, 331820068035, 1395291176641, 5327752138987, 18698405435444, 60922707883197, 185814239933254, 534246250634068, 1456622823771075
Offset: 0
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