A378173 Array read by antidiagonals: T(n,k) is the number of proper antichain partitions of the rectangular poset of size n X k.

1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 14, 38, 14, 1, 1, 42, 372, 372, 42, 1, 1, 132, 4282, 14606, 4282, 132, 1, 1, 429, 55149

Offset: 1

Author

Ludovic Schwob, Nov 18 2024

Keywords

• nonn
• tabl
• more

Comments

A partition of a poset into antichains is said to be proper if it does not contain two antichains A_1 and A_2, with x_1,y_1 in A_1 and x_2,y_2 in A_2, such that x_1y_2.

A proper antichain partition of a poset is endowed with an order relation, which is induced by the order relation of the poset. Let Y be a young diagram, and P the poset of shape Y. The number of linear extensions of P is the number of standard Young tableaux with shape Y. The sum over all proper antichain partitions of P, of the numbers of linear extensions of the induced orders, is equal to the number of normal generalized Young tableaux of shape Y with all rows and columns strictly increasing (cf. A299968).

Examples

			Array begins:
=====================================================================
n/k | 1     2      3      4      5      6 ...
----+----------------------------------------------------------------
  1 | 1     1      1      1      1      1 ...
  2 | 1     2      5     14     42    132 ...
  3 | 1     5     38    372   4282  55149 ...
  4 | 1    14    372  14606 ...
  5 | 1    42   4282 ...
  6 | 1   132  55149 ...
		

Crossrefs

Cf. A060854, A299968, A374985.

Formula

T(n,k) = T(k,n).

T(n,2) = A000108(n).