We introduce a new family of noncommutative analogs of the HallLittlewood symmetric functions. Our construction relies upon Tevlin’s bases and simple q-deformations of the classical combinatorial Hopf algebras. We connect our new Hall-Littlewood functions to permutation tableaux, and also give an exact formula for the q-enumeration of permutation tableaux of a fixed shape. This gives an explici...

A rectangulation is a tiling of a rectangle by a finite number of rectangles. The rectangulation is called generic if no four of its rectangles share a single corner. We initiate the enumeration of generic rectangulations up to combinatorial equivalence by establishing an explicit bijection between generic rectangulations and a set of permutations defined by a pattern-avoidance condition analog...

Enumeration, alias counting, is the oldest mathematical subject, while Algebraic Combinatorics is one of the youngest. Some cynics claim that Algebraic Combinatorics is not really a new subject but just a new name given to Enumerative Combinatorics in order to enhance its (former) poor image, but Algebraic Combinatorics is in fact the synthesis of two opposing trends: abstraction of the concret...

Our work studies the enumeration and random generation of unlabeled combinatorial classes of unrooted graphs. While the technique of vertex pointing provides a straightforward procedure for analyzing a labeled class of unrooted graphs by first studying its rooted counterpart, the existence of nontrivial symmetries in the unlabeled case causes this technique to break down. Instead, techniques su...

Recently the first author and Jang Soo Kim introduced lecture hall tableaux in their study of multivariate little q-Jacobi polynomials. They then enumerated bounded showed that enumeration is closely related to standard semistandard Young tableaux. In this paper we asymptotic behavior these thanks two other combinatorial models: non-intersecting paths on a graph whose faces are squares pentagon...

Combinatorial properties of zeons have been applied to graph enumeration problems, colorings, routing problems in communication networks, partition-dependent stochastic integrals, and Boolean satisfiability. Power series elementary zeon functions are naturally reduced finite sums by virtue the nilpotent zeons. Further, extension any analytic complex function has polynomial representations on as...

We study the enumeration problem for different kind of tree parking functions introduced recently, called functions, distributions, prime and rooted labelled trees important combinatorial families including ordered, unordered binary trees. Using decompositions underlying structures yields, after solving resulting equations, implicit characterizations suitable generating total number such size n...

In this paper, we discuss the computational complexity of the following enumeration problem: given a rational convex polyhedron P defined by a system of linear inequalities, output each vertex of P. It is still an open question whether there exists an algorithm for listing all vertices in running time polynomial in the input size and the output size. Informally speaking, a linear running time i...

In general, the representation of combinatorial objects is decisive for the feasibility of several enumerative tasks. In this work, we show how a (unique) string representation for (complete) initially-connected deterministic automata (ICDFA’s) with n states over an alphabet of k symbols can be used for counting, exact enumeration, sampling and optimal coding, not only the set of ICDFA’s but, t...

A permutation group G (acting on a set Ω, usually infinite) is said to be oligomorphic if G has only finitely many orbits on Ωn (the set of n-tuples of elements of Ω). Such groups have traditionally been linked with model theory and combinatorial enumeration; more recently their group-theoretic properties have been studied, and links with graded algebras, Ramsey theory, topological dynamics, an...