Enumerative Properties of NC^B(p,q)

I. P. Goulden; Alexandru Nica; Ion Oancea

Home/Archive/arXiv

arXiv · 0708.2212

Enumerative Properties of NC^B(p,q)

I. P. Goulden·Alexandru Nica·Ion Oancea

View original source ↗Read source document ↗Export metadata ↓

Abstract

We determine the rank generating function, the zeta polynomial and the Moebius function for the poset NC^B(p,q) of annular non-crossing partitions of type B, where p and q are two positive integers. We give an alternative treatment of some of these results in the case q=1, for which this poset is a lattice. We also consider the general case of multiannular non-crossing partitions of type B, and prove that this reduces to the cases of non-crossing partitions of type B in the annulus and in the disc.

Explore related subjects

math.CO math.GR

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

I. P. Goulden, Alexandru Nica, Ion Oancea. 2008-11-20. Enumerative Properties of NC^B(p,q). https://arxiv.org/abs/0708.2212

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Partitioning perfect graphs into comparability graphs

We study how many comparability subgraphs are needed to partition the edge set of a perfect graph. We show that many classes of perfect graphs can be partitioned into (at most) two comparability subgraphs and this holds for almost all perfect graphs. On the other hand, we prove that for interval graphs an arbitrarily large number of comparability subgraphs might be necessary.

math.CO↗

arXiv2026

On the homogeneity problem of the Kazhdan-Lusztig ideals

In this paper, our purpose is to find some techniques to identify inhomogeneous Kazhdan-Lusztig ideals. A set of generators of a Kazhdan-Lusztig ideal corresponds to some minors in some southwest submatrices inside some ambient matrix. We have defined a notion called a path inside a square matrix as a tool to work with determinants of the above mentioned minors. Using paths, we derive necessary and sufficient conditions for when such minors are inhomogeneous, singular, non-singular, and when there is divisibility between two terms of two such distinct minors. As a conclusion, we first provide an algorithm through which we can identify some sufficient conditions for a Kazhdan-Lusztig ideal to be inhomogeneous (by utilizing all the above mentioned results). Secondly, we provide certain sufficient conditions for a Kazhdan-Lusztig ideal to be standard homogeneous, and then we also prove that under certain assumptions, these conditions are also necessary for a Kazhdan-Lusztig ideal to be standard homogeneous.

math.CO↗

arXiv2026

Random Permutation Matrices Form a Basis with High Probability

Let $d=(n-1)^2+1$, the dimension of the real linear span of the $n\times n$ permutation matrices. We prove that $d$ independent uniformly random permutation matrices fail to form a basis with probability $(1+o(1))n^2(1-1/n)^d=(e^{3/2}+o(1))n^2e^{-n}$. The same asymptotic holds for a uniformly random $d$-element subset, confirming a conjecture of Kushwaha and Tripathi and identifying the leading obstruction: a matrix position avoided by every sample. More generally, for any fixed number of additional samples, we determine the first three exponential orders of the failure probability. To prove these results, we develop support estimates valid over arbitrary fields, derive Fourier bounds from permanental minors, and introduce a counting argument for concentrated assignment functionals over large prime fields. We also give a sparse lifting argument showing that real rank deficiency without an annihilating functional of small support has probability $o(e^{-Cn})$ for every fixed $C>0$.

math.CO↗