Enumeration of totally positive Grassmann cells

By Lauren K. Williams. 2003

151 citations recorded by the source corpus.

Postnikov (Webs in totally positive Grassmann cells, in preparation) has given a combinatorially explicit cell decomposition of the totally nonnegative part of a Grassmannian, denoted Grk,n+, and showed that this set of cells is isomorphic as a graded poset to many other interesting graded posets. The main result of our work is an explicit generating function which enumerates the cells in Grk,n+ according to their dimension. As a corollary, we give a new proof that the Euler characteristic of Grk,n+ is 1. Additionally, we use our result to produce a new q-analog of the Eulerian numbers, which interpolates between the Eulerian numbers, the Narayana numbers, and the binomial coefficients.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/cd0664625efc42ff01669be4a7a89c0176dce98e

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Citations in the corpus

Total positivity for cominuscule Grassmannians (2007, background): “6 Enumeration of cells The cells in the totally non-negative Grassmannian for type A were enumerated in (15; 8).”
Soliton Solutions of the KP Equation and Application to Shallow Water Waves (2009, methodology): “It turns out that the classification problem is closely related to the characterization of the totally nonnegative Grassmannian cells of Gr(N,M) recently studied by Postnikov et al (see for example [29, 34]).”
KP solitons in shallow water (2010, methodology): “This question has been solved by Postnikov and his colleagues (see [42, 51]), and our classification theorem of the soliton solutions provides an alternative proof based on a simple asymptotic analysis as described in section 5 (see also [6, 8]).”
Matching polytopes, toric geometry, and the totally non-negative Grassmannian (2009, background): “Other combinatorial aspects of (Grk,n)≥0, and more generally of (G/P)≥0, were investigated by Marsh and Rietsch [12], Rietsch [16], and the third author [ 22 , 23].”
Formulae for Askey-Wilson moments and enumeration of staircase tableaux (2010, background): “ δ = q = 0, the generating function for the numbers Zn(y) = Zn(y;α,β,γ,δ;q) is X∞ n=0 Zn(y)wn = (1+t)(1+yt) where w = t (1+t)(1+yt) . Note that this is the generating function of the Narayana numbers [32]. Corollary 3.10. When α = β = γ = 1 and δ = q = 0, the generating function for the numbers Zn(y”
Generalized permutation patterns - a short survey (2008, background): “This is obviously true also for the patterns 21 and 2 1, since the only permutation with no descent and no inversion is the increasing permutation 123 . . . n.”
Scattering Amplitudes and the Positive Grassmannian (2012, background): “Starting with the generic configuration in G+(2, 4), we find the boundaries defined by ∂ given as follows [105]:”
Shelling totally nonnegative flag varieties (2005, background): “When G=PJ is the (type A) Grassmannian, Q J is the poset of cells of the totally nonnegative Grassmannian, studied by Postnikov [18] and the author [27].”
Total positivity, Grassmannians, and networks (2006, background): “lternating strand diagrams. We also construct bijections between Γ-diagramsand other combinatorialobjects such as permutations of a certain kind, rook placements on skew Young diagrams, etc. Williams [W1], Steingrimsson-Williams [SW], and Corteel-Williams [CW] obtained several enumerative results on”