Related papers: A bijection between unicellular and bicellular map…
For a directed graph G on vertices {0,1,...,n}, a G-parking function is an n-tuple (b_1,...,b_n) of non-negative integers such that, for every non-empty subset U of {1,...,n}, there exists a vertex j in U for which there are more than b_j…
To comprehend the multipartite organization of large-scale biological and social systems, we introduce a new information theoretic approach that reveals community structure in weighted and directed networks. The method decomposes a network…
We give a bijective correspondence between the number of nilpotent matrices over a Boolean semiring and the number of directed acyclic graphs on ordered vertices. We then enumerate pairs of maps between two finite sets whose composites are…
We give another bijective proof of a result of Corteel and Nadeau. We find a generating function related to unrestricted columns of permutation tableaux. As a consequence, we obtain a sign-imbalance formula for permutation tableaux. We…
Various biological phenomena, like cell differentiation and pattern formation in multicellular organisms, are explained using the bifurcation theory. Molecular network motifs like positive feedback and mutual repressor exhibit bifurcation…
We explain how homotopical information of two composeable relations can be organized in two simplicial categories that augment the relations row and column complexes. We show that both of these categories realize to weakly equivalent…
Rapid advance of experimental techniques provides an unprecedented in-depth view into complex developmental processes. Still, little is known on how the complexity of multicellular organisms evolved by elaborating developmental programs and…
In the present paper we study the geometry of the closed Bia{\l}ynicki-Birula cells of the quiver Grassmannians associated to a nilpotent representation of a cyclic quiver defined by a single matrix. For the special case, where we choose…
The two tableaux assigned by the Robinson--Schensted correspondence are equal if and only if the input permutation is an involution, so the RS algorithm restricts to a bijection between involutions in the symmetric group and standard…
Phylogenetic trees are binary nonplanar trees with labelled leaves, and plane oriented recursive trees are planar trees with an increasing labelling. Both families are enumerated by double factorials. A bijection is constructed, using the…
We present a bijective algorithm with which an arbitrary permutation decomposes canonically into elementary blocks which we call families, which are sets with a specified number of ascents and descents. We show that families, arranged in an…
We describe birational representations of discrete groups generated by involutions, having their origin in the theory of exactly solvable vertex-models in lattice statistical mechanics. These involutions correspond respectively to two kinds…
Maps are polygonal cellular networks on Riemann surfaces. This paper analyzes the construction of closed form general representations for the enumerative generating functions associated to maps of fixed but arbitrary genus. The method of…
We give a recursion formula to generate all equivalence classes of biconnected graphs with coefficients given by the inverses of the orders of their groups of automorphisms. We give a linear map to produce a connected graph with say, u,…
A planar stuffed map is an embedding of a graph into the 2-sphere $S^{2}$, considered up to orientation-preserving homeomorphisms, such that the complement of the graph is a collection of disjoint topologically connected components that are…
Metrics for indirectly predicting the folding rates of RNA sequences are of interest. In this letter, we introduce a simple metric of RNA structural complexity, which accounts for differences in the energetic contributions of RNA base…
A unicellular collection on a surface is a collection of curves whose complement is a single disk. There is a natural surgery operation on unicellular collections, endowing the set of such with a graph structure where the edge relation is…
We present a bijection from planar reduced trees to planar rooted hypertrees, which extends Knuth's rotation correspondence between planar binary trees and planar rooted trees. The operadic counterpart of the new bijection is explained.…
The polytope structure of the associahedron is decomposed into two categories, types and classes. The classification of types is related to integer partitions, whereas the classes present a new combinatorial problem. We solve this and…
Closed-form generating functions for counting one-face rooted hypermaps with a known number of darts by number of vertices and edges is found, using matrix integral expressions relating to the reduced density operator of a bipartite quantum…