A uniform bijection between nonnesting and noncrossing partitions

Abstract

Abstract. In 2007, D.I. Panyushev defined a remarkable map on the set of nonnesting partitions (antichains in the root poset of a finite Weyl group). In this paper we identify Panyushev's map with the Kreweras complement on the set of noncrossing partitions, and hence construct the first uniform bijection between nonnesting and noncrossing partitions. Unfortunately, the proof that our construction is well-defined is case-by-case, using a computer in the exceptional types. Fortunately, the proof involves new and interesting combinatorics in the classical types. As consequences, we prove several conjectural properties of the Panyushev map, and we prove two cyclic sieving phenomena conjectured by D. Bessis and V. Reiner.

Brick polytopes of spherical subword complexes and generalized associahedra

“…The downward projection used to construct the quotient associates to any element w ∈ W the maximal c-sortable element below w. It is denoted by π c ↓ , see [ between c-sortable elements and facets of the subword complex SC(cw • (c)) is the restriction to c-sortable elements of the surjective map κ from W to facets as defined in Section 5. 1. It moreover yields the connection between κ and the map π c ↓ .…”

“…• in [29], the authors further study EL-labelings and canonical spanning trees for subword complexes, with applications to generation and order theoretic properties of the subword complex, and discuss their relation to alternative EL labelings of Cambrian lattices in [19], • in [30], the authors show that the vertex barycenter of generalized associahedra coincide with that of their corresponding permutahedra, using the vertex description of generalized associahedra presented here, • in [6], the first author and C. Ceballos provide a combinatorial description of the denominator vectors based on the subword complex approach, • in his recent dissertation [43], N. Williams provides an amazing conjecture together with a huge amount of computational evidence that the present approach to cluster complexes and generalized associahedra as well yields a type-independent and explicit bijection to nonnesting partitions (see also [1] for further background on this connection).…”

“…The c ex -sorting word of w • is w • (c ex ) = τ 1 τ 2 τ 3 τ 1 τ 2 τ 1 . We thus consider the word c ex w • (c ex ) := τ 1 τ 2 τ 3 τ 1 τ 2 τ 3 τ 1 τ 2 τ1 . The facets of the subword complex SC(c ex w • (c ex )) are {1, 2, 3}, {2, 3, 4}, {3, 4, 5}, {4, 5, 6}, {5, 6, 7}, {6, 7, 8}, {6, 8, 9}, {4, 6, 9}, {3, 5, 7}, {2, 4, 9}, {1, 3, 7}, {1, 7, 8}, {1, 8, 9}, and {1, 2, 9}.…”

Noncrossing partitions and Bruhat order

“…For variety, we use the Kreweras complement defined on both noncrossing and nonnesting partitions to prove this, in a manner very similar in spirit to the ideas in [2] and [20]. Note that this will also follow from Theorem 4.15, which shows that D = V.…”

“…Consider the noncrossing partition π = (1, 6)(2, 3, 5) from figure 2. Then m π = s [1,2] s [2,4] s [1,6] …”