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 consider the poset of weighted partitions Π w n , introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of Π w n provide a generalization of the lattice Π n of partitions, which we show possesses many of the well-known properties of Π n . In particular, we prove these intervals are EL-shellable, we show that the Möbius invariant of each maximal interval is given up to sign by the number of rooted trees on node set {1, 2, . . . , n} having a fixed number of descents, we find combinatorial bases for homology and cohomology, and we give an explicit sign twisted S n -module isomorphism from cohomology to the multilinear component of the free Lie algebra with two compatible brackets. We also show that the characteristic polynomial of Π w n has a nice factorization analogous to that of Π n .

The Catalan numbers constitute one of the most important sequences in combinatorics. Catalan objects have been generalized in various directions, including the classical Fuss-Catalan objects and the rational Catalan generalization of Armstrong-Rhoades-Williams. We propose a wider generalization of these families indexed by a composition s which is motivated by the combinatorics of planar rooted trees; when s = (2, ..., 2) and s = (k + 1, ..., k + 1) we recover the classical Catalan and Fuss-Catalan combinatorics, respectively. Furthermore, to each pair (a, b) of relatively prime numbers we can associate a signature that recovers the combinatorics of rational Catalan objects. We present explicit bijections between the resulting s-Catalan objects, and a fundamental recurrence that generalizes the fundamental recurrence of the classical Catalan numbers. Our framework allows us to define signature generalizations of parking functions which coincide with the generalized parking functions studied by Pitman-Stanley and Yan, as well as generalizations of permutations which coincide with the notion of Stirling multipermutations introduced by Gessel-Stanley. Some of our constructions differ from the ones of Armstrong-Rhoades-Williams, however as a byproduct of our extension, we obtain the additional notions of rational permutations and rational trees.Contents arXiv:1805.03863v1 [math.CO] 10 May 2018 40References 40 1 a+b a+b b , also known as the rational Catalan number. When a = n and b = n + 1 we recover the classical Catalan numbers, and when a = n and b = kn + 1 we obtain another classical generalization known as the Fuss-Catalan numbers. Even thought ingredients of the rational Catalan story have appeared previously in the literature, Armstrong, Rhodes and Williams started a more systematic study of the rational Catalan objects C a,b in [3]. In this generalization of the classical story there are rational versions P a,b of the parking

Available online xxxx MSC: 05E45 05E18 05A18 17B01 18D50It is a classical result that the multilinear component of the free Lie algebra is isomorphic (as a representation of the symmetric group) to the top (co)homology of the proper part of the poset of partitions Π n tensored with the sign representation. We generalize this result in order to study the multilinear component of the free Lie algebra with multiple compatible Lie brackets. We introduce a new poset of weighted partitions Π k n that allows us to generalize the result. The new poset is a generalization of Π n and of the poset of weighted partitions Π w n introduced by Dotsenko and Khoroshkin and studied by the author and Wachs for the case of two compatible brackets. We prove that the poset Π k n with a top element added is EL-shellable and hence CohenMacaulay. This and other properties of Π k n enable us to answer questions posed by Liu on free multibracketed Lie algebras. In particular, we obtain various dimension formulas and multicolored generalizations of the classical Lyndon and comb bases for the multilinear component of the free Lie algebra. We also obtain a plethystic formula for the Frobenius characteristic of the representation of the symmetric group on the multilinear component of the free multibracketed Lie algebra.

A multivariate polynomial is stable if it is non-vanishing whenever all variables have positive imaginary parts. A matroid has the weak half-plane property (WHPP) if there exists a stable polynomial with support equal to the set of bases of the matroid. If the polynomial can be chosen with all of its nonzero coefficients equal to one then the matroid has the half-plane property (HPP). We describe a systematic method that allows us to reduce the WHPP to the HPP for large families of matroids. This method makes use of the Tutte group of a matroid. We prove that no projective geometry has the WHPP and that a binary matroid has the WHPP if and only if it is regular. We also prove that T_8 and R_9 fail to have the WHPP.Comment: 8 pages. To appear in J. Combin. Theory Ser.