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 A056941 Number of antichains (or order ideals) in the poset 5*m*n or plane partitions with not more than m rows, n columns 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 J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy] P. A. MacMahon, Combinatory analysis. FORMULA From Peter Bala, Oct 13 2011: (Start) Product_{k=0..4} C(n+m+k, m+k)/C(n+k, k) 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 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) Determinants of 5 X 5 subarrays of Pascal's triangle A007318 (a matrix entry being set to 0 when not present). Also determinants of 5 X 5 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 EXAMPLE The array starts: [1    1      1        1          1           1            1 ...] [1    6     21       56        126         252          462 ...] [1   21    196     1176       5292       19404        60984 ...] [1   56   1176    14112     116424      731808      3737448 ...] [1  126   5292   116424    1646568    16818516    133613766 ...] [1  252  19404   731808   16818516   267227532   3184461423 ...] [1  462  60984  3737448  133613766  3184461423  55197331332 ...] [...] PROG (PARI) A056941(n, m)=prod(k=0, 4, binomial(n+m+k, m+k)/binomial(n+k, k) \\ M. F. Hasler, Sep 26 2018 CROSSREFS Cf. A000372, A056932, A001263, A056939, A056940. Antidiagonals sum to A005363 (Hoggatt sequence). Sequence in context: A060972 A144066 A296827 * A157638 A142596 A176063 Adjacent sequences:  A056938 A056939 A056940 * A056942 A056943 A056944 KEYWORD nonn,easy,tabl AUTHOR EXTENSIONS Edited by M. F. Hasler, Sep 26 2018 STATUS approved

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Last modified January 18 23:25 EST 2021. Contains 340262 sequences. (Running on oeis4.)