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
Keywords
References
- Richard P. Stanley: Enumerative Combinatorics, vol. 2, p. 378.
Links
- Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 25.
- 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