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

0,5

Comments

Triangle of generalized binomial coefficients (n,k)_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

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

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)

Example

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

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 *)

Prog

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

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.

Sequence in context: A136267 A109960 A196019 * A168288 A157523 A141691

Adjacent sequences: A056937 A056938 A056939 * A056941 A056942 A056943

Keyword

nonn,easy,tabl

Author

Mitch Harris

Extensions

Edited by M. F. Hasler, Sep 26 2018

Status

approved