

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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


LINKS



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 welldefined as a formal power series) det(A_{ij}), i,j > 0, where A_{11} = 0, A_{1j} = Sum_{k=1..j1} x^(k(jk)) if j > 1, A_{i1} = 1 if i > 1, A_{ii} = 1 if i > 1, A_{ij} = x^(i(ji)) 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



KEYWORD

easy,nonn


AUTHOR



STATUS

approved



