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Register a new userThree interacting families of Fuss-Catalan posets
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Three families of posets depending on a nonnegative integer parameter $m$ are introduced. The underlying sets of these posets are enumerated by the $m$-Fuss Catalan numbers. Among these, one is a generalization of Stanley lattices and another one is a generalization of Tamari lattices. The three families of posets are related: they fit into a chain for the order extension relation and they share some properties. Two associative algebras are constructed as quotients of generalizations of the Malvenuto-Reutenauer algebra. Their products describe intervals of our analogues of Stanley lattices and Tamari lattices. In particular, one is a generalization of the Loday-Ronco algebra.
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We introduce $delta$-cliffs, a generalization of permutations and increasing trees depending on a range map $delta$. We define a first lattice structure on these objects and we establish general results about its subposets. Among them, we describe sufficient conditions to have EL-shellable posets, lattices with algorithms to compute the meet and the join of two elements, and lattices constructible by interval doubling. Some of these subposets admit natural geometric realizations. Then, we introduce three families of subposets which, for some maps $delta$, have underlying sets enumerated by the Fuss-Catalan numbers. Among these, one is a generalization of Stanley lattices and another one is a generalization of Tamari lattices. These three families of posets fit into a chain for the order extension relation and they share some properties. Finally, in the same way as the product of the Malvenuto-Reutenauer algebra forms intervals of the right weak order of permutations, we construct algebras whose products form intervals of the lattices of $delta$-cliff. We provide necessary and sufficient conditions on $delta$ to have associative, finitely presented, or free algebras. We end this work by using the previous Fuss-Catalan posets to define quotients of our algebras of $delta$-cliffs. In particular, one is a generalization of the Loday-Ronco algebra and we get new generalizations of this structure.
The main objects of the present paper are (i) Hibi rings (toric rings arising from order polytopes of posets), (ii) stable set rings (toric rings arising from stable set polytopes of perfect graphs), and (iii) edge rings (toric rings arising from edge polytopes of graphs satisfying the odd cycle condition). The goal of the present paper is to analyze those three toric rings and to discuss their structures in the case where their class groups have small rank. We prove that the class groups of (i), (ii) and (iii) are torsionfree. More precisely, we give descriptions of their class groups. Moreover, we characterize the posets or graphs whose associated toric rings have rank $1$ or $2$. By using those characterizations, we discuss the differences of isomorphic classes of those toric rings with small class groups.
A combinatorial structure, $mathcal{F}$, with counting sequence ${a_n}_{nge 0}$ and ordinary generating function $G_mathcal{F}=sum_{nge0} a_n x^n$, is positive algebraic if $G_mathcal{F}$ satisfies a polynomial equation $G_mathcal{F}=sum_{k=0}^N p_k(x),G_mathcal{F}^k $ and $p_k(x)$ is a polynomial in $x$ with non-negative integer coefficients. We show that every such family is associated with a normed $mathbf{n}$-magma. An $mathbf{n}$-magma with $mathbf{n}=(n_1,dots, n_k)$ is a pair $mathcal{M}$ and $mathcal{F}$ where $mathcal{M}$ is a set of combinatorial structures and $mathcal{F}$ is a tuple of $n_i$-ary maps $f_i,:,mathcal{M}^{n_i}to mathcal{M}$. A norm is a super-additive size map $||cdot||,:, mathcal{M}to mathbb{N} $. If the normed $mathbf{n}$-magma is free then we show there exists a recursive, norm preserving, universal bijection between all positive algebraic families $mathcal{F}_i$ with the same counting sequence. A free $mathbf{n}$-magma is defined using a universal mapping principle. We state a theorem which provides a combinatorial method of proving if a particular $mathbf{n}$-magma is free. We illustrate this by defining several $mathbf{n}$-magmas: eleven $(1,1)$-magmas (the Fibonacci families), seventeen $(1,2)$-magmas (nine Motzkin and eight Schroder families) and seven $(3)$-magmas (the Fuss-Catalan families). We prove they are all free and hence obtain a universal bijection for each $mathbf{n}$. We also show how the $mathbf{n}$-magma structure manifests as an embedded bijection.
Squared singular values of a product of s square random Ginibre matrices are asymptotically characterized by probability distribution P_s(x), such that their moments are equal to the Fuss-Catalan numbers or order s. We find a representation of the Fuss--Catalan distributions P_s(x) in terms of a combination of s hypergeometric functions of the type sF_{s-1}. The explicit formula derived here is exact for an arbitrary positive integer s and for s=1 it reduces to the Marchenko--Pastur distribution. Using similar techniques, involving Mellin transform and the Meijer G-function, we find exact expressions for the Raney probability distributions, the moments of which are given by a two parameter generalization of the Fuss-Catalan numbers. These distributions can also be considered as a two parameter generalization of the Wigner semicircle law.
Motivated by generalizing Khovanovs categorification of the Jones polynomial, we study functors $F$ from thin posets $P$ to abelian categories $mathcal{A}$. Such functors $F$ produce cohomology theories $H^*(P,mathcal{A},F)$. We find that CW posets, that is, face posets of regular CW complexes, satisfy conditions making them particularly suitable for the construction of such cohomology theories. We consider a category of tuples $(P,mathcal{A},F,c)$, where $c$ is a certain ${1,-1}$-coloring of the cover relations in $P$, and show the cohomology arising from a tuple $(P,mathcal{A},F,c)$ is functorial, and independent of the coloring $c$ up to natural isomorphism. Such a construction provides a framework for the categorification of a variety of familiar topological/combinatorial invariants: anything expressible as a rank-alternating sum over a thin poset.
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