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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.

%I A056941 #61 Nov 15 2022 02:41:36
%S A056941 1,1,1,1,6,1,1,21,21,1,1,56,196,56,1,1,126,1176,1176,126,1,1,252,5292,
%T A056941 14112,5292,252,1,1,462,19404,116424,116424,19404,462,1,1,792,60984,
%U A056941 731808,1646568,731808,60984,792,1,1,1287,169884,3737448,16818516,16818516,3737448,169884,1287,1
%N 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.
%C A056941 Triangle of generalized binomial coefficients (n,k)_5; cf. A342889. - _N. J. A. Sloane_, Apr 03 2021
%D A056941 Berman and Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), p. 103-124
%D A056941 P. A. MacMahon, Combinatory Analysis, Section 495, 1916.
%D A056941 R. P. Stanley, Theory and application of plane partitions. II. Studies in Appl. Math. 50 (1971), p. 259-279. Thm. 18.1
%H A056941 Seiichi Manyama, <a href="/A056941/b056941.txt">Rows n = 0..139 of triangle, flattened</a>
%H A056941 J. Berman and P. Koehler, <a href="/A006356/a006356.pdf">Cardinalities of finite distributive lattices</a>, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
%H A056941 Johann Cigler, <a href="https://arxiv.org/abs/2103.01652">Pascal triangle, Hoggatt matrices, and analogous constructions</a>, arXiv:2103.01652 [math.CO], 2021.
%H A056941 Johann Cigler, <a href="https://www.researchgate.net/publication/349376205_Some_observations_about_Hoggatt_triangles">Some observations about Hoggatt triangles</a>, Universität Wien (Austria, 2021).
%H A056941 P. A. MacMahon, <a href="http://name.umdl.umich.edu/ABU9009.0001.001">Combinatory analysis</a>.
%H A056941 <a href="/index/Pos#posets">Index entries for sequences related to posets</a>
%F A056941 From _Peter Bala_, Oct 13 2011: (Start)
%F A056941 A(n, k) = Product_{j=0..4} C(n+k+j, k+j)/C(n+j, j) gives the array as a square.
%F A056941 g(n-1, k-1)*g(n, k+1)*g(n+1, k) = g(n-1, k)*g(n, k-1)*g(n+1, k+1) where g(n, k) is the array A(n, k) and triangle T(n, k).
%F A056941 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)
%F A056941 From _Peter Bala_, May 10 2012: (Start)
%F A056941 Determinants of 5 X 5 subarrays of Pascal's triangle A007318 (a matrix entry being set to 0 when not present).
%F A056941 Also determinants of 5 X 5 arrays whose entries come from a single row:
%F A056941   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)]. (End)
%F A056941 From _G. C. Greubel_, Nov 14 2022: (Start)
%F A056941 T(n, k) = Product_{j=0..4} binomial(n+j, k)/binomial(k+j, k) (gives the triangle).
%F A056941 Sum_{k=0..n} T(n, k) = A005363(n). (End)
%e A056941 The array starts:
%e A056941   [1    1      1        1          1           1            1 ...]
%e A056941   [1    6     21       56        126         252          462 ...]
%e A056941   [1   21    196     1176       5292       19404        60984 ...]
%e A056941   [1   56   1176    14112     116424      731808      3737448 ...]
%e A056941   [1  126   5292   116424    1646568    16818516    133613766 ...]
%e A056941   [1  252  19404   731808   16818516   267227532   3184461423 ...]
%e A056941   [1  462  60984  3737448  133613766  3184461423  55197331332 ...]
%e A056941   [...]
%e A056941 Considered as a triangle, the initial rows are:
%e A056941    1;
%e A056941    1,   1;
%e A056941    1,   6,     1;
%e A056941    1,  21,    21,      1;
%e A056941    1,  56,   196,     56,       1;
%e A056941    1, 126,  1176,   1176,     126,      1;
%e A056941    1, 252,  5292,  14112,    5292,    252,     1;
%e A056941    1, 462, 19404, 116424,  116424,  19404,   462,   1;
%e A056941    1, 792, 60984, 731808, 1646568, 731808, 60984, 792, 1; ...
%t A056941 T[n_, k_] := Product[Binomial[n+j, k]/Binomial[k+j, k], {j,0,4}];
%t A056941 Table[T[n, k], {n,0,13}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 14 2022 *)
%o A056941 (PARI) A056941(n,m)=prod(k=0,4,binomial(n+m+k,m+k)/binomial(n+k,k)) \\ as an array \\ _M. F. Hasler_, Sep 26 2018
%o A056941 (Magma)
%o A056941 A056941:= func< n,k | (&*[Binomial(n+j,k)/Binomial(k+j,k): j in [0..4]]) >;
%o A056941 [A056941(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 14 2022
%o A056941 (SageMath)
%o A056941 def A056941(n,k): return product(binomial(n+j,k)/binomial(k+j,k) for j in (0..4))
%o A056941 flatten([[A056941(n,k) for k in range(n+1)] for n in range(14)]) # _G. C. Greubel_, Nov 14 2022
%Y A056941 Cf. A000372, A001263, A005363 (row sums), A056932, A056939, A056940, A142465, A142467, A142468.
%Y A056941 Antidiagonals sum to A005363 (Hoggatt sequence).
%Y A056941 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.
%K A056941 nonn,easy,tabl
%O A056941 0,5
%A A056941 _Mitch Harris_
%E A056941 Edited by _M. F. Hasler_, Sep 26 2018