arXiv subjects

Bruce E Sagan

Publications and source records attributed to Bruce E Sagan.

3 recordsLinked to original sources

arXiv2026

Ordered set partition posets

A set partition is said to be ordered if the blocks of the partition are listed in a specific order. The ordered set partitions of $\{1,\ldots,n\}$, with a unique minimal element adjoined, form a lattice $\Om_n$ with respect to refinement. The lattice $\Om_n$ is well known to be the face lattice of the permutohedron. In this paper we study the combinatorics and topology of two subposets of $\Om_n$ with restricted block sizes, either all divisible by some fixed $d\ge2$, or all congruent to $1$ modulo $d$. For the $d$-divisible case we derive an explicit recursive atom ordering for the lattice, as well as formulas for the action of the symmetric group on the Whitney homology and the rank-selected homology, and also for the multiplicity of the trivial representation. In the 1 mod $d$ case we show that the poset has a curious interval structure related to the $k$-Catalan numbers. Our investigations lead to enumerative invariants in both cases. Open problems and avenues for future research are scattered throughout.

math.CO↗

arXiv2024

Stirling numbers for complex reflection groups

In an earlier paper, we defined and studied q-analogues of the Stirling numbers of both types for the Coxeter group of type B. In the present work, we show how this approach can be extended to all irreducible complex reflection groups G. The Stirling numbers of the first and second kind are defined via the Whitney numbers of the first and second kind, respectively, of the intersection lattice of G. For the groups G(m,p,n), these numbers and polynomials can be given combinatorial interpretations in terms of various statistics. The ordered version of ths q-Stirling numbers of the second kind also show up in conjectured Hilbert series for certain super coinvariant algebras.

math.CO↗

arXiv2021

The Amazing Chromatic Polynomial

Let G be a combinatorial graph with vertices V and edges E. A proper coloring of G is an assignment of colors to the vertices such that no edge connects two vertices of the same color. These are the colorings considered in the famous Four Color Theorem. It turns out that the number of proper colorings of G using t colors is a polynomial in t, called the chromatic polynomial of G. This polynomial has many wonderful properties. It also has the surprising habit of appearing in contexts which, a priori, have nothing to do with graph coloring. We will survey three such instances involving acyclic orientations, hyperplane arrangements, and increasing forests. In addition, connections to symmetric functions and algebraic geometry will be mentioned.

math.CO↗