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SUMMARY:Digjoy Paul (Tata Institute of Fundamental Research)
DTSTART:20211028T060000Z
DTEND:20211028T070000Z
DTSTAMP:20260423T004639Z
UID:ARCSIN/5
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/ARCSIN/5/">S
 ymmetric $q\,t$ Catalan polynomials</a>\nby Digjoy Paul (Tata Institute of
  Fundamental Research) as part of ARCSIN - Algebra\, Representations\, Com
 binatorics and Symmetric functions in INdia\n\n\nAbstract\nThe $q\, t$-Cat
 alan functions $C_n(q\,t)$\, an $q\, t$- analogue of Catalan numbers\, wer
 e first introduced in connection with Macdonald polynomials and Garsia–H
 aiman’s theory of diagonal harmonics [1996] as certain rational function
 s in $q$ and $t$. Haglund [2003] and shortly after that\, Haiman announced
  two combinatorial interpretations of $C_n(q\,t)$ as a weighted sum over a
 ll Dyck paths. An open problem related to these polynomials is a combinato
 rial proof of its symmetry in $q$ and $t$.\n\nWe define two symmetric $q\,
 t$ Catalan polynomials on Dyck paths and provide proof of the symmetry by 
 establishing an involution on plane trees. We also give a combinatorial pr
 oof of a result by Garsia et al. regarding parking functions and the numbe
 r of connected graphs. This is joint work with Joseph Pappe and Anne Schil
 ling.\n
LOCATION:https://researchseminars.org/talk/ARCSIN/5/
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