A097994 T(n,k) counts plane partitions of n that can be 'extended' in (k+2) ways to a plane partition of n+1 by adding 1 element to it. Equivalently, it counts how many partitions of n have (k+2) different partitions of n+1 just covering it.

1, 3, 0, 3, 3, 0, 6, 6, 0, 1, 3, 15, 3, 3, 0, 9, 21, 6, 12, 0, 0, 3, 34, 21, 25, 3, 0, 0, 10, 45, 36, 54, 15, 0, 0, 0, 6, 54, 72, 108, 36, 6, 0, 0, 0, 9, 84, 102, 172, 117, 15, 0, 1, 0, 0, 3, 84, 174, 306, 228, 54, 7, 3, 0, 0, 0, 18, 114, 225, 483, 447, 162, 18, 12, 0, 0, 0, 0, 3, 114

Offset: 1

Author

Wouter Meeussen, Sep 07 2004

Keywords

• hard
• nonn
• tabl

Comments

The first column starts a t k=3 since all plane partitions can be extended in at least 3 ways. Row sums are A000219 by definition. Sum T(n,k) (k+2) =A090984.

Examples

			T(4,4)=1 because {{2,1},{1}} is the only plane partition of 4 that can be extended in 4+2 = 6 ways to a plane partition of 5.
		

Crossrefs

Cf. A000219, A090984.

Programs

Mathematica

(* functions 'planepartitions' and 'coversplaneQ', see A096574 *) Table[Frequencies[Count[planepartitions[n+1], q_/; coversplaneQ[q, # ]]&/@ planepartitions[n]], {n, 1, 16}]