## Lectures

Seminar information archive ～10/10｜Next seminar｜Future seminars 10/11～

### 2012/07/23

16:30-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Combinatorial Ergodicity (ENGLISH)

**Thomas W. Roby**(University of Connecticut)Combinatorial Ergodicity (ENGLISH)

[ Abstract ]

Many cyclic actions $\\tau$ on a finite set $S$ of

combinatorial objects, along with many natural

statistics $\\phi$ on $S$, exhibit``combinatorial ergodicity'':

the average of $\\phi$ over each $\\tau$-orbit in $S$ is

the same as the average of $\\phi$ over the whole set $S$.

One example is the case where $S$ is the set of

length $n$ binary strings $a_{1}\\dots a_{n}$

with exactly $k$ 1's,

$\\tau$ is the map that cyclically rotates them,

and $\\phi$ is the number of \\textit{inversions}

(i.e, pairs $(a_{i},a_{j})=(1,0)$ for $iJ$ less than $j$).

This phenomenon was first noticed by Panyushev

in 2007 in the context of antichains in root posets;

Armstrong, Stump, and Thomas proved his

conjecture in 2011.

We describe a theoretical framework for results of this kind,

and discuss old and new results for products of two chains.

This is joint work with Jim Propp.

Many cyclic actions $\\tau$ on a finite set $S$ of

combinatorial objects, along with many natural

statistics $\\phi$ on $S$, exhibit``combinatorial ergodicity'':

the average of $\\phi$ over each $\\tau$-orbit in $S$ is

the same as the average of $\\phi$ over the whole set $S$.

One example is the case where $S$ is the set of

length $n$ binary strings $a_{1}\\dots a_{n}$

with exactly $k$ 1's,

$\\tau$ is the map that cyclically rotates them,

and $\\phi$ is the number of \\textit{inversions}

(i.e, pairs $(a_{i},a_{j})=(1,0)$ for $iJ$ less than $j$).

This phenomenon was first noticed by Panyushev

in 2007 in the context of antichains in root posets;

Armstrong, Stump, and Thomas proved his

conjecture in 2011.

We describe a theoretical framework for results of this kind,

and discuss old and new results for products of two chains.

This is joint work with Jim Propp.