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

0,5

Comments

Triangle of generalized binomial coefficients (n,k)_5; cf. A342889. - N. J. A. Sloane, Apr 03 2021

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

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)

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 ...]
  [...]
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; ...

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 *)

Prog

(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
(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
(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

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.

Sequence in context: A144066 A363849 A296827 * A157638 A347975 A142596

Adjacent sequences: A056938 A056939 A056940 * A056942 A056943 A056944

Keyword

nonn,easy,tabl

Author

Mitch Harris

Extensions

Edited by M. F. Hasler, Sep 26 2018

Status

approved