A098529 Triangle read by rows: T(n,k) counts plane partitions of n+1 that can be 'shrunk' in k ways to a plane partition of n by removing 1 element from it. Equivalently, it counts how many partitions of n+1 have k different partitions of n it just covers.

1, 3, 3, 3, 6, 6, 1, 3, 18, 3, 9, 24, 15, 3, 42, 38, 3, 10, 60, 69, 21, 6, 72, 153, 45, 6, 9, 114, 220, 141, 15, 1, 3, 120, 399, 274, 60, 3, 18, 159, 558, 570, 162, 12, 3, 174, 834, 1029, 399, 46, 9, 267, 1080, 1749, 921, 138, 3

Offset: 0

Author

Wouter Meeussen, Sep 12 2004

Keywords

• more
• nonn
• tabf

Comments

Sequence starts 1; 3; 3,3; 6,6,1; 3,18,3; 9,24,15; 3,42,38,3; Row sums are A000219= the plane partitions of n+1 apart from offset. Sum(all k, k * T(n,k) ) = A090984(n) by definition. First column is A007425. Row lengths are A120565. - Franklin T. Adams-Watters, Jun 14 2006

Examples

			T(4,1)=2 because the only plane partitions of 4+1=5 that can be shrunk in only 1 way to plane partitions of 4 are {{5}} and {{1},{1},{1},{1},{1}}, producing {{4}} and {{1},{1},{1},{1}} respectively.
T(4,1)=3 because the only plane partitions of 4+1=5 that can be shrunk in only 1 way to plane partitions of 4 are {{5}},{{1,1,1,1,1}} and {{1},{1},{1},{1},{1}}, producing {{4}},{{1,1,1,1}} and {{1},{1},{1},{1}} respectively.
		

Crossrefs

Cf. A000219, A090984, A007425, A120565.

Programs

Mathematica

(* functions 'planepartitions' and 'coversplaneQ', see A096574 *) Table[Frequencies[Count[planepartitions[n], q_/; coversplaneQ[ #, q]]&/@ planepartitions[n+1]], {n, 1, 12}]

Extensions

Corrected and extended by Franklin T. Adams-Watters, Jun 14 2006

More terms from Wouter Meeussen, May 05 2007