1.3k citations · 3k across the 62 of their papers we have counts for
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Bessenrodt-Stanley polynomials and the octahedron recurrence
Philippe Di Francesco
We show that a family of multivariate polynomials recently introduced by Bessenrodt and Stanley can be expressed as solution of the octahedron recurrence with suitable initial data…
A doubly-refined enumeration of alternating sign matrices and descending plane partitions
Roger E. Behrend, Philippe Di Francesco, Paul Zinn-Justin
It was shown recently by the authors that, for any n, there is equality between the distributions of certain triplets of statistics on nxn alternating sign matrices (ASMs) and desc…
On the weighted enumeration of alternating sign matrices and descending plane partitions
Roger E. Behrend, Philippe Di Francesco, Paul Zinn-Justin
We prove a conjecture of Mills, Robbins and Rumsey [Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (1983), 340-359] that, for any n, k, m an…
The solution of the T-system for arbitrary boundary
P. Di Francesco
We present an explicit solution of the -system for arbitrary boundary conditions. For each boundary, this is done by constructing a network, i.e. a graph with positively w…
Q-system Cluster Algebras, Paths and Total Positivity
Philippe Di Francesco, Rinat Kedem
In the first part of this paper, we provide a concise review of our method of solution of the Q-systems in terms of the partition function of paths on a weighted graph. In th…
Q-systems, Heaps, Paths and Cluster Positivity
P. Di Francesco, R. Kedem
We consider the cluster algebra associated to the -system for as a tool for relating -system solutions to all possible sets of initial data. We show that the conserved…
Integrability of graph combinatorics via random walks and heaps of dimers
P. Di Francesco, E. Guitter
We investigate the integrability of the discrete non-linear equation governing the dependence on geodesic distance of planar graphs with inner vertices of even valences. This equat…