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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.
0
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
Atli Fannar Franklín, Pattern avoidance enumerated by inversions, arXiv:2410.07467 [math.CO], 2024. See pp. 2, 13-14.
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 * A377173 A081165 A289749
KEYWORD
easy,nonn,more
AUTHOR
Richard Stanley, Oct 04 2006
STATUS
approved