A117629 Number of Gorenstein partitions of n, i.e., those partitions of n whose corresponding Schubert variety has a Gorenstein homogeneous coordinate ring, or equivalently those partitions of n which, when regarded as order ideals of PxP (where P={1,2,...}), have all maximal chains of the same length.

1, 2, 3, 3, 5, 5, 5, 7, 10, 5, 11, 11, 11, 15, 15, 8, 23, 19, 21, 21, 27, 15, 29, 39, 34, 36, 37, 22, 61, 37

Offset

1,2

Links

Table of n, a(n) for n=1..30.

Atli Fannar Franklín, Anders Claesson, Christian Bean, Henning Úlfarsson, and Jay Pantone, Restricted Permutations Enumerated by Inversions, arXiv:2406.16403 [cs.DM], 2024. See p. 4.

R. P. Stanley, Hilbert functions of graded algebras, Advances in Math. 28 (1978), 57-83 (Theorem 5.4).

T. Svanes, Coherent cohomology of Schubert subschemes of flag schemes and applications, Advances in Math. 14 (1974), 369-453 (Theorem 5.5.6).

Formula

f(n) is the number of finite sequences of length > 1 of positive integers such that n is the second elementary symmetric function of the terms of the sequence. The ordinary generating function for f(n) is the infinite determinant (which is well-defined as a formal power series) det(A_{ij}), i,j > 0, where A_{11} = 0, A_{1j} = -Sum_{k=1..j-1} x^(k(j-k)) if j > 1, A_{i1} = 1 if i > 1, A_{ii} = 1 if i > 1, A_{ij} = -x^(i(j-i)) if j > i > 1 and A_{ij} = 0 if i > j > 1.

Example

f(10)=5 because the Gorenstein partitions of 10 are (10), (5,5), (2,2,2,2,2), (1,1,1,1,1,1,1,1,1,1) and (4,3,2,1). The sequences for which 10 is the second elementary symmetric function are (1,10), (2,5), (5,2), (10,1) and (1,1,1,1,1).

Crossrefs

Sequence in context: A086162 A036703 A306253 * A081165 A289749 A087172

Adjacent sequences: A117626 A117627 A117628 * A117630 A117631 A117632

Keyword

easy,nonn,more

Author

Richard Stanley, Oct 04 2006

Status

approved