Proper Mergings of Stars and Chains are Counted by Sums of Antidiagonals in Certain Convolution Arrays -- The Details

Abstract

A proper merging of two disjoint quasi-ordered sets $P$ and $Q$ is a quasi-order on the union of $P$ and $Q$ such that the restriction to $P$ or $Q$ yields the original quasi-order again and such that no elements of $P$ and $Q$ are identified. In this article, we determine the number of proper mergings in the case where $P$ is a star (i.e. an antichain with a smallest element adjoined), and $Q$ is a chain. We show that the lattice of proper mergings of an $m$-antichain and an $n$-chain, previously investigated by the author, is a quotient lattice of the lattice of proper mergings of an $m$-star and an $n$-chain, and we determine the number of proper mergings of an $m$-star and an $n$-chain by counting the number of congruence classes and by determining their cardinalities. Additionally, we compute the number of Galois connections between certain modified Boolean lattices and chains.

arXiv comment27 pages, 7 figures, 1 table. Jonathan Farley has solved Problem 4.18; added Section 4.4 to describe his solution

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• Proper Mergings of Stars and Chains are Counted by Sums of Antidiagonals in Certain Convolution Arrays -- The Details

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• Proper Mergings of Stars and Chains are Counted by Sums of Antidiagonals in Certain Convolution Arrays -- The Details