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Lectures
Seminar information archive ～12/26｜Next seminar｜Future seminars 12/27～
2012/07/23
16:30-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Thomas W. Roby (University of Connecticut)
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.
