A140901 Number of 3 X 5 matrices with elements in 0..n with each row and each column in nondecreasing order. 3,5,n can be permuted, see formula.

1, 56, 1176, 14112, 116424, 731808, 3737448, 16195608, 61408347, 208416208, 644195552, 1837984512, 4892876352, 12259074816, 29115302688, 65937597264, 143107211709, 298915373064, 603074875480, 1178943365600, 2239226847000, 4142127132000, 7477931097000

Offset

0,2

References

Richard P. Stanley: Enumerative Combinatorics, vol. 2, p. 378.

Links

Table of n, a(n) for n=0..22.

Grigory M., Number of matrices with weakly increasing rows and columns, MathStackExchange.

W. F. Wheatley and James Ethridge (Proposers), Comment from Alan H. Rapoport, Problem 84, Missouri Journal of Mathematical Sciences, volume 8, #2, Spring 1996, pages 97-102.

Formula

(Empirical) Set p,q,r to n,5,3 (in any order) in s=p+q+r-1; a(n) = Product_{i=0..r-1} (binomial(s,p+i)*i!/(s-i)^(r-i-1)).

(Conjecture) G.f.: (1 + 40*x + 400*x^2 + 1456*x^3 + 2212*x^4 + 1456*x^5 + 400*x^6 + 40*x^7 + x^8)/(1-x)^16. - Bruno Berselli, May 08 2012

a(n) = Product_{i=1..3} Product_{j=1..5} Product_{k=1..n+1} (i + j + k - 1) / (i + j + k - 2). See the section on plane partitions with bounded part size in Stanley's reference. This comment is relevant to the sequences A140902 - A140943 as well. - Sela Fried, Oct 18 2023

Crossrefs

Cf. A140902-A140943.

Sequence in context: A219583 A341429 A002009 * A160346 A219371 A183587

Adjacent sequences: A140898 A140899 A140900 * A140902 A140903 A140904

Keyword

nonn

Author

R. H. Hardin, Jul 05 2008

Status

approved