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UID:www.tcs.tifr.res.in/event/948
DTSTAMP:20230914T125944Z
SUMMARY:The Combinatorics of Cyclic Orders
DESCRIPTION:Speaker: Arvind Ayyer (Department of Mathematics\nIndian Instit
ute of Science\nBangalore)\n\nAbstract: \nAbstract: A cyclic order on a s
et X is a ternary relation on X satisfying three conditions called cyclici
ty\, asymmetry and transitivity. These can be thought of as analogues of p
artial orders. There are many similarities between posets and cyclically o
rdered sets. In particular\, there is a notion of a circular extension and
there is a natural generalisations of the order polytope defined by Stanl
ey. Cyclic orders were recently introduced in the combinatorial literature
to enumerate descent classes of permutations.\nWe introduce several class
es of polytopes contained in $[0\,1]^n$ and cut out by inequalities involv
ing sums of consecutive coordinates. We show that the normalized volumes o
f these polytopes enumerate circular extensions of certain partial cyclic
orders. Among other things this gives a new point of view on a question po
pularized by Stanley (Exercise~4.56(d) in Enumerative combinatorics\, Vol
ume 1). We also provide a combinatorial interpretation of the Ehrhart h^*-
polynomials of some of these polytopes in terms of descents of total cycli
c orders. The Euler numbers\, the Eulerian numbers and the Narayana number
s appear as special cases (a joint work with M. Josuat-Verg\\`es and S. Ra
massamy).\n
URL:https://www.tcs.tifr.res.in/web/events/948
DTSTART;TZID=Asia/Kolkata:20190311T113000
DTEND;TZID=Asia/Kolkata:20190311T123000
LOCATION:A-201 (STCS Seminar Room)
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