A Computer-Generated Theorem by Shalosh B. Ekhad
Consider the poset L:= , [{}, {1}, {1}, {1, 2, 3}]
Let P[n] be the poset obtained from joining n copies
of L together, by identifying the last, 1, vertices
of the previous link to the first, 1, vertices of the next link
Let , F[n](x), be the weight-enumerator of
of all P-partitions of P[n], with the weight q^(sum of entries)
a
times the product of , x
where , a, are the last , 1, entries
of the P-partition
F[n](x), satisfies the following
functional-recurrence equation:
4 2 3
(q x - 1) F[n - 1](q x)
F[n](x) = - --------------------------------
2 2 3
(q x - 1) (q x - 1) (q x - 1)
Subject to the intial condition
4 2
q x - 1
F[1](x) = -------------------------------------------
2 2 3 4
(q x - 1) (q x - 1) (q x - 1) (q x - 1)
The pure generating function for all P-partitions is
F[n](1)
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