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, 2, vertices
of the previous link to the first, 2, vertices of the next link
Let , F[n](x, y), be the weight-enumerator of
of all P-partitions of P[n], with the weight q^(sum of entries)
a b
times the product of , x y
where , a, b, are the last , 2, entries
of the P-partition
F[n](x, y), satisfies the following
functional-recurrence equation:
2
q x F[n - 1](1, q x y) F[n - 1](q x, q y)
F[n](x, y) = - ----------------------- + -------------------
2 (q y - 1) (q x - 1)
(q x y - 1) (q x - 1)
Subject to the intial condition
4 2
q y x - 1
F[1](x, y) = -----------------------------------------------------------
2 3 4 2
(q y - 1) (q y - 1) (q y x - 1) (q x y - 1) (q x y - 1)
The pure generating function for all P-partitions is
F[n](1, 1)
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