Adler V.E. On the combinatorics of several integrable hierarchies, talk at The Mini-Workshop on Integrable Equations, 17 February 2015, Independent University of Moscow (abstract)
Speaker: Vsevolod Adler
Title: On the combinatorics of several integrable hierarchies
We establish a correspondence between the generating functions related to some integrable hierarchies and certain classes of set partitions. In a sense, the inverse problem of enumerative combinatorics is considered: given a prescribed statistics, to figure out a definition of the corresponding combinatorial objects. The answer for the potential Burgers hierarchy is rather well known: set partitions, Bell polynomials, Stirling numbers of the 2nd kind, Bell numbers. An analogous correspondence for the KdV hierarchy (non-overlapping partitions, Bessel numbers) and Ibragimov–Shabat hierarchy (B type partitions, Dowling numbers) is new.
Slides: http://matphys.itp.ac.ru/talks/Coefficients_talk_MCCME_2015_02_17.pdf, local copy
http://matphys.itp.ac.ru/talks/Coefficients_talk_ITP_2015_01_16.pdf, local copy (in Russian)
Adler V.E. On the combinatorics of several integrable hierarchies, arXiv:1501.06086