ABSTRACT. An n-core partition is an integer partition whose Young diagram contains no hook lengths equal to n. We consider partitions that are simultaneously a-core and b-core for two relatively prime integers a and b. These are related to abacus diagrams and the combinatorics of the affine symmetric group (type A). We observe that self-conjugate simultaneous core partitions correspond to the combinatorics of type C, and use abacus diagrams to unite the discussion of these two sets of objects.In particular, we prove that 2n-and (2mn + 1)-core partitions correspond naturally to dominant alcoves in the m-Shi arrangement of type Cn, generalizing a result of Fishel-Vazirani for type A. We also introduce a major index statistic on simultaneous n-and (n + 1)-core partitions and on self-conjugate simultaneous 2n-and (2n + 1)-core partitions that yield q-analogues of the Coxeter-Catalan numbers of type A and type C.We present related conjectures and open questions on the average size of a simultaneous core partition, q-analogs of generalized Catalan numbers, and generalizations to other Coxeter groups. We also discuss connections with the cyclic sieving phenomenon and q, t-Catalan numbers.
We introduce new families of combinatorial objects whose enumeration computes volumes of flow polytopes. These objects provide an interpretation, based on parking functions, of Baldoni and Vergne's generalization of a volume formula originally due to Lidskii. We recover known flow polytope volume formulas and prove new volume formulas for flow polytopes. A highlight of our model is an elegant formula for the flow polytope of a graph we call the caracol graph.As by-products of our work, we uncover a new triangle of numbers that interpolates between Catalan numbers and the number of parking functions, we prove the log-concavity of rows of this triangle along with other sequences derived from volume computations, and we introduce a new Ehrhart-like polynomial for flow polytope volume and conjecture product formulas for the polytopes we consider.Dedicated to the memory of Griff L. Bilbro.
We give a generating function for the fully commutative affine permutations enumerated by rank and Coxeter length, extending formulas due to Stembridge and Barcucci-Del Lungo-Pergola-Pinzani. For fixed rank, the length generating functions have coefficients that are periodic with period dividing the rank. In the course of proving these formulas, we obtain results that elucidate the structure of the fully commutative affine permutations.2000 Mathematics Subject Classification. Primary 05A15, 05E15, 20F55; Secondary 05A30.
For systems containing large numbers of ions, calculations using Density Functional Theory (DFT) are often impractical because of the amount of time needed to perform the computations. In this paper, we show that weighted-average Madelung constants of MgO nanotubes correlate in an essentially perfectly linear way with cohesive energies determined by DFT. We discuss this correlation in terms of the relationship between lattice energies and cohesive energies. Through this linear correlation, Madelung constants are used to predict cohesive energies and average ion charges of nanostructures containing up to 3940 ions. Cohesive energies of MgO nanotubes are shown to converge to a value lower than those of bulk MgO. Using the slopes of the DFT versus Madelung constant plots, the average charges on the ions in the nanotubes are determined. For nanotubes containing the same number of ions, the relative stability of longer tubes versus disc-like structures is discussed.
By means of the Ehrhart theory of inside-out polytopes we establish a general counting theory for nonattacking placements of chess pieces with unbounded straight-line moves, such as the queen, on a polygonal convex board. The number of ways to place q identical nonattacking pieces on a board of variable size n but fixed shape is given by a quasipolynomial function of n, of degree 2q, whose coefficients are polynomials in q. The number of combinatorially distinct types of nonattacking configuration is the evaluation of our quasipolynomial at n = −1. The quasipolynomial has an exact formula that depends on a matroid of weighted graphs, which is in turn determined by incidence properties of lines in the real affine plane. We study the highest-degree coefficients and also the period of the quasipolynomial, which is needed if the quasipolynomial is to be interpolated from data, and which is bounded by some function, not well understood, of the board and the piece's move directions.In subsequent parts we specialize to the square board and then to subsets of the queen's moves, and we prove exact formulas (most but not all already known empirically) for small numbers of queens, bishops, and nightriders.Each part concludes with open questions, both specialized and broad.
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