A056940 Number of antichains (or order ideals) in the poset 4*m*n or plane partitions with at most m rows and n columns and entries <= 4.

1, 1, 1, 1, 5, 1, 1, 15, 15, 1, 1, 35, 105, 35, 1, 1, 70, 490, 490, 70, 1, 1, 126, 1764, 4116, 1764, 126, 1, 1, 210, 5292, 24696, 24696, 5292, 210, 1, 1, 330, 13860, 116424, 232848, 116424, 13860, 330, 1, 1, 495, 32670, 457380, 1646568, 1646568, 457380, 32670, 495, 1

Offset: 0

Author

Mitch Harris

Keywords

• nonn
• easy
• tabl

Comments

Triangle of generalized binomial coefficients (n,k)A342889.%20-%20_N.%20J.%20A.%20Sloane">4; cf. A342889. - _N. J. A. Sloane, Apr 03 2021

Determinants of 4 X 4 subarrays of Pascal's triangle A007318 (a matrix entry being set to 0 when not present). - Gerald McGarvey, Feb 24 2005

Row sums are: {1, 2, 7, 32, 177, 1122, 7898, 60398, 494078, 4274228, 38763298, ...}. - Roger L. Bagula, Mar 08 2010

Also determinants of 4x4 arrays whose entries come from a single row: T(n,k) = det [C(n,k), C(n,k-1), C(n,k-2), C(n,k-3); C(n,k+1), C(n,k), C(n,k-1), C(n,k-2); C(n,k+2), C(n,k+1), C(n,k), C(n,k-1); C(n,k+3), C(n,k+2), C(n,k+1), C(n,k)]. - Peter Bala, May 10 2012

Examples

			Triangle begins as:
  1.
  1,   1.
  1,   5,     1.
  1,  15,    15,      1.
  1,  35,   105,     35,      1.
  1,  70,   490,    490,     70,      1.
  1, 126,  1764,   4116,   1764,    126,     1.
  1, 210,  5292,  24696,  24696,   5292,   210,   1.
  1, 330, 13860, 116424, 232848, 116424, 13860, 330, 1. - _Roger L. Bagula_, Mar 08 2010
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened
• Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
• J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
• Johann Cigler, Pascal triangle, Hoggatt matrices, and analogous constructions, arXiv:2103.01652 [math.CO], 2021.
• Johann Cigler, Some observations about Hoggatt triangles, Universität Wien (Austria, 2021).
• P. A. MacMahon, Combinatory analysis, sect. 495, 1916.
• R. P. Stanley, Theory and application of plane partitions, II. Studies in Appl. Math. 50 (1971), p. 259-279. DOI:10.1002/sapm1971503259. Thm. 18.1.
• Index entries for sequences related to posets

Crossrefs

Cf. A000372, A056932, A001263.

Antidiagonals sum to A005362 (Hoggatt sequence).

Cf. A056939 (q=2), A056940 (q=3), A056941 (q=4), A142465 (q=5), A142467 (q=6), A142468 (q=7), A174109 (q=8).

Triangles of generalized binomial coefficients (n,k)_m (or generalized Pascal triangles) for m = 1,...,12: A007318 (Pascal), A001263, A056939, A056940, A056941, A142465, A142467, A142468, A174109, A342889, A342890, A342891.

Programs

Mathematica

c[n_, q_] = Product[i + j, {j, 0, q}, {i, 1, n}];
T[n_, m_, q_] = c[n, q]/(c[m, q]*c[n - m, q]);
Table[T[n, k, 3], {n, 0, 10}, {k, 0, n}]//Flatten (* Roger L. Bagula, Mar 08 2010 *)(* modified by G. C. Greubel, Apr 13 2019 *)

PARI

A056940(n,m)=prod(k=0,3,binomial(n+m+k,m+k)/binomial(n+k,k)) \\ M. F. Hasler, Sep 26 2018

Formula

Product_{k=0..3} C(n+m+k, m+k)/C(n+k, k) gives the array as a square.

T(n,m,q) = c(n,q)/(c(m,q)*c(n-m,q)) with c(n,q) = Product_{i=1..n, j=0..q} (i + j), q = 3. - Roger L. Bagula, Mar 08 2010

From Peter Bala, Oct 13 2011: (Start)

T(n-1,k-1)*T(n,k+1)*T(n+1,k) = T(n-1,k)*T(n,k-1)*T(n+1,k+1).

Define f(r,n) = n!*(n+1)!*...*(n+r)!. The triangle whose (n,k)-th entry is f(r,0)*f(r,n)/(f(r,k)*f(r,n-k)) is A007318 (r = 0), A001263 (r = 1), A056939 (r = 2), A056940 (r = 3) and A056941 (r = 4). (End)

Extensions

Edited by M. F. Hasler, Sep 26 2018