
TL;DR
This paper introduces a geometric and combinatorial framework for analyzing West's stack-sorting map using discrete convexity and polyhedral geometry, revealing new structural insights and conjectures.
Contribution
It develops the concept of fertilitopes as nestohedra to characterize the fertility of permutations and connects these structures to cumulant formulas and stack-sorting properties.
Findings
Fertilitopes are nestohedra derived from binary trees.
Set of fertility numbers has density 1 among natural numbers.
All infertility numbers up to size 126 are classified.
Abstract
We introduce tools from discrete convexity theory and polyhedral geometry into the theory of West's stack-sorting map . Associated to each permutation is a particular set of integer compositions that appears in a formula for the fertility of , which is defined to be . 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 is a transversal discrete polymatroid when it is nonempty. We define the fertilitope of to be the convex hull of , 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…
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Taxonomy
TopicsStochastic processes and statistical mechanics · Advanced Combinatorial Mathematics · Random Matrices and Applications
