Parallel projection of matroid spheres
Parallelepipeds, nilpotent groups and Gowers norms
In his proof of Szemerédi’s Theorem, Gowers introduced certain norms that are defined on a parallelepiped structure. A natural question is on which sets a parallelepiped structure (and thus a Gowers norm) can be defined. We focus on dimensions and and show when this possible, and describe a correspondence between the parallelepiped structures and nilpotent groups.
Parallelism in diagram geometry.
Parameter augmentation for two formulas.
Parametric production matrices and weighted succession rules: a Dyck path example.
Parastrophically Equivalent Quasigroup Equations
Parity indices of partitions
Parity of numbers of crossings for complete -partite graphs
Parity of the partition function.
Parity results for broken k-diamond partitions and (2k+1)-cores
Parity systems and the delta-matroid intersection problem.
Parity theorems for statistics on domino arrangements.
Parity theorems for statistics on lattice paths and Laguerre configurations.
Parity theorems for statistics on permutations and Catalan words.
Parity versions of 2-connectedness.
Parity vertex colorings of binomial trees
We show for every k ≥ 1 that the binomial tree of order 3k has a vertex-coloring with 2k+1 colors such that every path contains some color odd number of times. This disproves a conjecture from [1] asserting that for every tree T the minimal number of colors in a such coloring of T is at least the vertex ranking number of T minus one.
Parity vertex colouring of graphs
A parity path in a vertex colouring of a graph is a path along which each colour is used an even number of times. Let χₚ(G) be the least number of colours in a proper vertex colouring of G having no parity path. It is proved that for any graph G we have the following tight bounds χ(G) ≤ χₚ(G) ≤ |V(G)|-α(G)+1, where χ(G) and α(G) are the chromatic number and the independence number of G, respectively. The bounds are improved for trees. Namely, if T is a tree with diameter diam(T) and radius rad(T),...
Parity-alternating permutations and successions
The study of parity-alternating permutations of {1, 2, … n} is extended to permutations containing a prescribed number of parity successions - adjacent pairs of elements of the same parity. Several enumeration formulae are computed for permutations containing a given number of parity successions, in conjunction with further parity and length restrictions. The objects are classified using direct construction and elementary combinatorial techniques. Analogous results are derived for circular permutations....
Parking functions and noncrossing partitions.
Parking functions, empirical processes, and the width of rooted labeled trees.