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.

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, 16818516, 3737448, 169884, 1287, 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">5; cf. A342889. - _N. J. A. Sloane, Apr 03 2021

Examples

			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 ...]
  [...]
Considered as a triangle, the initial rows are:
   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; ...
		

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

• Seiichi Manyama, Rows n = 0..139 of triangle, flattened
• 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.
• Index entries for sequences related to posets

Crossrefs

Cf. A000372, A001263, A005363 (row sums), A056932, A056939, A056940, A142465, A142467, A142468.

Antidiagonals sum to A005363 (Hoggatt sequence).

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

Magma

A056941:= func< n,k | (&*[Binomial(n+j,k)/Binomial(k+j,k): j in [0..4]]) >;
[A056941(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 14 2022

Mathematica

T[n_, k_] := Product[Binomial[n+j, k]/Binomial[k+j, k], {j,0,4}];
Table[T[n, k], {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 14 2022 *)

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

SageMath

def A056941(n,k): return product(binomial(n+j,k)/binomial(k+j,k) for j in (0..4))
flatten([[A056941(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Nov 14 2022

Formula

From Peter Bala, Oct 13 2011: (Start)

A(n, k) = Product_{j=0..4} C(n+k+j, k+j)/C(n+j, j) gives the array as a square.

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).

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)

From Peter Bala, May 10 2012: (Start)

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)]. (End)

From G. C. Greubel, Nov 14 2022: (Start)

T(n, k) = Product_{j=0..4} binomial(n+j, k)/binomial(k+j, k) (gives the triangle).

Sum_{k=0..n} T(n, k) = A005363(n). (End)

Extensions

Edited by M. F. Hasler, Sep 26 2018