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A056941 Antichains (or order ideals) in the poset 5*m*n or plane partitions with rows <= m, columns <= n and entries <= 5 11
1, 1, 1, 1, 6, 1, 1, 21, 21, 1, 1, 56, 196, 56, 1, 1, 126, 1176, 1176, 126, 1, 1, 252, 5292, 14112, 5292, 252, 1, 1, 462, 19404, 116424, 116424, 19404, 462, 1, 1, 792, 60984, 731808, 1646568, 731808, 60984, 792, 1, 1, 1287, 169884, 3737448, 16818516 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

REFERENCES

Berman and Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), p. 103-124

P. A. MacMahon, Combinatory Analysis, Section 495, 1916.

R. P. Stanley, Theory and application of plane partitions. II. Studies in Appl. Math. 50 (1971), p. 259-279. Thm. 18.1

LINKS

Table of n, a(n) for n=0..49.

P. A. MacMahon, Combinatory analysis.

Index entries for sequences related to posets

FORMULA

From Peter Bala, Oct 13 2011: (Start)

Product[ C(n+m+k, m+k)/C(n+k, k), {k, 0, 4} ] gives the array as a square.

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

Determinants of 5x5 subarrays of Pascal's triangle A007318 (a matrix entry being set to 0 when not present).

Also determinants of 5x5 arrays whose entries come from a single row:

  det [C(n,k),C(n,k-1),C(n,k-2),C(n,k-3),C(n,k-4); C(n,k+1),C(n,k),C(n,k-1),C(n,k-2),C(n,k-3); C(n,k+2),C(n,k+1),C(n,k),C(n,k-1),C(n,k-2); C(n,k+3),C(n,k+2),C(n,k+1),C(n,k),C(n,k-1); C(n,k+4),C(n,k+3),C(n,k+2),C(n,k+1),C(n,k)]. - Peter Bala, May 10 2012

CROSSREFS

Cf. A000372, A056932, A001263, A056939, A056940.

Antidiagonals sum to A005363 (Hoggatt sequence)

Sequence in context: A203954 A060972 A144066 * A157638 A142596 A176063

Adjacent sequences:  A056938 A056939 A056940 * A056942 A056943 A056944

KEYWORD

nonn,easy,tabl

AUTHOR

Mitch Harris

STATUS

approved

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Last modified April 16 06:19 EDT 2014. Contains 240545 sequences.