Arithmetic of marked order polytopes, monotone triangle reciprocity, and partial colorings

For a poset P, a subposet A, and an order preserving map F from A into the real numbers, the marked order polytope parametrizes the order preserving extensions of F to P. We show that the function counting integral-valued extensions is a piecewise polynomial in F and we prove a reciprocity statement in terms of order-reversing maps. We apply our results to give a geometric proof of a combinatorial reciprocity for monotone triangles due to Fischer and Riegler (2011) and we consider the enumerative problem of counting extensions of partial graph colorings of Herzberg and Murty (2007).Comment: 17 pages, 10 figures; V2: minor changes (including title); V3: examples included (suggested by referee), to appear in "SIAM Journal on Discrete Mathematics

Order ideals in weak subposets of Young's lattice and associated unimodality conjectures

The k-Young lattice Y^k is a weak subposet of the Young lattice containing partitions whose first part is bounded by an integer k>0. The Y^k poset was introduced in connection with generalized Schur functions and later shown to be isomorphic to the weak order on the quotient of the affine symmetric group by a maximal parabolic subgroup. We prove a number of properties for $Y^k$ including that the covering relation is preserved when elements are translated by rectangular partitions with hook-length $k$. We highlight the order ideal generated by an $m\times n$ rectangular shape. This order ideal, L^k(m,n), reduces to L(m,n) for large k, and we prove it is isomorphic to the induced subposet of L(m,n) whose vertex set is restricted to elements with no more than k-m+1 parts smaller than m. We provide explicit formulas for the number of elements and the rank-generating function of L^k(m,n). We conclude with unimodality conjectures involving q-binomial coefficients and discuss how implications connect to recent work on sieved q-binomial coefficients.Comment: 18 pages, 5 figure

On the homomorphism order of labeled posets

Partially ordered sets labeled with k labels (k-posets) and their homomorphisms are examined. We give a representation of directed graphs by k-posets; this provides a new proof of the universality of the homomorphism order of k-posets. This universal order is a distributive lattice. We investigate some other properties, namely the infinite distributivity, the computation of infinite suprema and infima, and the complexity of certain decision problems involving the homomorphism order of k-posets. Sublattices are also examined.Comment: 14 page