CanaDAM 2013 Memorial University of Newfoundland, June 10 - 13, 2013 www.cms.math.ca//2013
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Enumerative Combinatorics
Organizer and Chair: Marni Mishna (Simon Fraser University)
[PDF]

MATHILDE BOUVEL, LaBRI, CNRS and Univ. Bordeaux (France)
Operators of equivalent sorting power and related Wilf-equivalences  [PDF] [SLIDES]

We study partial sorting operators $\mathbf{A}$ on permutations that are obtained composing Knuth's stack sorting operator $\mathbf{S}$ and the reverse operator $\mathbf{R}$, as many times as desired. For any such operator $\mathbf{A}$, we provide a statistics-preserving bijection between the set of permutations sorted by $\mathbf{S} \circ \mathbf{A}$ and the set of those sorted by $\mathbf{S} \circ \mathbf{R} \circ \mathbf{A}$. This is based on an apparently novel bijection between permutations avoiding the pattern $231$ and those avoiding $132$ which preserves many permutation statistics and has unexpected consequences in terms of Wilf-equivalences.

Joint work with Michael Albert (University of Otago, New Zealand).

SERGI ELIZALDE, Dartmouth College
Bijections for lattice paths between two boundaries  [PDF] [SLIDES]

We prove that on the set of lattice paths with north and east (unit) steps that lie between two boundaries $B$ and $T$, the statistics number of east steps shared with $B$' and number of east steps shared with $T$' have a symmetric joint distribution. We give an involution that switches these statistics, and a generalization to paths that contain south steps. We show that a similar result relates to the Tutte polynomial of a matroid. Finally, we extend our main theorem to $k$-tuples of paths, providing connections to flagged SSYT and $k$-triangulations. This is joint work with Martin Rubey.

Counting matrices over finite fields with zeroes on Rothe diagrams  [PDF] [SLIDES]

A $q$-analogue of permutations with restricted positions is invertible matrices over a finite field of size $q$ with support that avoids some entries. The number of such matrices may not be a polynomial in $q$ (Stembridge) but for some nice cases the numbers are nice polynomials. We generalize a result of Haglund by showing that when the support lies in a skew shape, the number of such matrices is a polynomial with nonnegative coefficients. We also study the situation when the zeroes are the entries of the Rothe diagram of a permutation.

Joint work with Aaron Klein and Joel Lewis.

MARKUS NEBEL, Kaiserslautern University
The Combinatorics of RNA in the Polymere Zeta Model  [PDF] [SLIDES]

Recently it has been observed that the computational prediction of RNA secondary structure can be speed-up by a linear factor on average. To this end, one has to assume the so-called polymere zeta property, i.e. two building blocks of an RNA molecule at distance $m$ are paired (in folding) with probability $b/m^c$, for some constants $b,c>0$. In this talk, we examine the averaged shape of an RNA folding in a polymere zeta model using generating functions and techniques form complex analysis. We find that some important structural motifs show a rather different behavior than observed in real world molecules.

BRUCE SAGAN, Michigan State University
A factorization theorem for $m$-rook placements  [PDF] [SLIDES]

Consider a Ferrers diagram $B=(b_1,b_2,\ldots,b_n)$. Let $r_k(B)$ be the number of placements of $k$ non-attacking rooks on $B$ and $x\hspace{-2pt}\downarrow_k=(x)(x-1)\cdots(x-k+1)$. The famous Factorization Theorem of Goldman-Joichi-White states that $\sum_{k\ge0} r_k(B) x\hspace{-2pt}\downarrow_{n-k}=\prod_j (x+b_j-j+1)$. Briggs and Remmel considered a generalization of rook placements to $m$-rook placements which are related to wreath products $C_m\wr S_N$ where $C_m$ is a cyclic group and $S_N$ a symmetric group. They were able to prove a version of the Factorization Theorem in this setting, but only for certain $B$. We give a generalization which holds for all $B$. This is joint work with Loehr and Remmel.