Fertilitopes

Abstract

We introduce tools from discrete convexity theory and polyhedral geometry into the theory of West's stack-sorting map s. Associated to each permutation π is a particular set V(π) of integer compositions that appears in a formula for the fertility of π, which is defined to be |s −1 (π)|. These compositions also feature prominently in more general formulas involving families of colored binary plane trees called troupes and in a formula that converts from free to classical cumulants in noncommutative probability theory. We show that V(π) is a transversal discrete polymatroid when it is nonempty. We define the fertilitope of π to be the convex hull of V(π), and we prove a surprisingly simple characterization of fertilitopes as nestohedra arising from full binary plane trees. Using known facts about nestohedra, we provide a procedure for describing the structure of the fertilitope of π directly from π using Bousquet-Mélou's notion of the canonical tree of π. As a byproduct, we obtain a new combinatorial cumulant conversion formula in terms of generalizations of canonical trees that we call quasicanonical trees. We also apply our results on fertilitopes to study combinatorial properties of the stack-sorting map. In particular, we show that the set of fertility numbers has density 1, and we determine all infertility numbers of size at most 126. Finally, we reformulate the conjecture that σ∈s −1 (π) x des(σ)+1 is always real-rooted in terms of nestohedra, and we propose natural ways in which this new version of the conjecture could be extended.

Stack-sorting for Coxeter groups

“…Much of the author's work on the stack-sorting map has relied on a certain Decomposition Lemma, which provides a recursive method for computing the fertility of a permutation, and a certain Fertility Formula, which gives an explicit expression for the fertility of a permutation as a sum over combinatorial objects called valid hook configurations. These tools have led to several new results about the stack-sorting map, including the aforementioned recurrence for counting 3-stack-sortable permutations [22], theorems about uniquely sorted permutations [29,21,42,60], a very surprising and useful connection with cumulants in noncommutative probability theory [28], and connections with certain polytopes called nestohedra [23]. In Section 8, we introduce affine valid hook configurations, and we prove analogues of the Decomposition Lemma and the Fertility Formula for affine permutations.…”

“…If δ = n , then a decreasing δ-permutree is the same thing as a decreasing binary plane tree, and the postorder reading defined here agrees with the standard postorder reading (see [10,25,28]). One definition of the stack-sorting map, which is responsible for much of the structure underlying it (such as its connection with free probability theory [28] and nestohedra [23]), combines the in-order reading with the postorder reading. More precisely, the stack-sorting map is given by s = P •I −1 n (see [10,25,28]).…”

“…This map was originally defined by West [63] as a deterministic variant of a stack-sorting machine introduced by Knuth [38]. West's stack-sorting map, which we will often simply call the stack-sorting map, has now received vigorous attention and has found connections with several other parts of combinatorics [10,12,11,13,28,22,21,25,23,29,35,42,60,64,33]. The second motivation for our terminology comes from the fact that when ≡ des is the descent congruence on W = S n , the map S ≡ des is the pop-stack-sorting map.…”

Stack-Sorting for Coxeter Groups

“…Much of the author's work on the stack-sorting map has relied on a certain Decomposition Lemma, which provides a recursive method for computing the fertility of a permutation, and a certain Fertility Formula, which gives an explicit expression for the fertility of a permutation as a sum over combinatorial objects called valid hook configurations. These tools have led to several new results about the stack-sorting map, including the aforementioned recurrence for counting 3stack-sortable permutations [22], theorems about uniquely sorted permutations [21,29,42,59], a very surprising and useful connection with cumulants in noncommutative probability theory [28], and connections with certain polytopes called nestohedra [23]. In Section 8, we introduce affine valid hook configurations, and we prove analogues of the Decomposition Lemma and the Fertility Formula for affine permutations.…”

“…If δ = n , then a decreasing δ-permutree is the same thing as a decreasing binary plane tree, and the postorder reading defined here agrees with the standard postorder reading (see [10,25,28]). One definition of the stack-sorting map, which is responsible for much of the structure underlying it (such as its connection with free probability theory [28] and nestohedra [23]), combines the in-order reading with the postorder reading. More precisely, the stack-sorting map is given by s = P • I −1 n (see [10,25,28]).…”