A solution to one of Knuth's permutation problems

We answer a problem posed recently by Knuth: an n-dimensional box, with edges lying on the positive coordinate axes and generic edge lengths W_1 < W_2 < ... < W_n, is dissected into n! pieces along the planes x_i = x_j. We describe which pieces have the same volume, and show that there are C_n distinct volumes, where C_n denotes the nth Catalan number.Comment: 4 pages, 2 figures

Computing a pyramid partition generating function with dimer shuffling

We verify a recent conjecture of Kenyon/Szendroi, arXiv:0705.3419, by computing the generating function for pyramid partitions. Pyramid partitions are closely related to Aztec Diamonds; their generating function turns out to be the partition function for the Donaldson--Thomas theory of a non-commutative resolution of the conifold singularity {x1x2 -x3x4 = 0}. The proof does not require algebraic geometry; it uses a modified version of the domino shuffling algorithm of Elkies, Kuperberg, Larsen and Propp.Comment: 19 pages, 13 figures. v2: fixed minor typos, updated references and future work; added some definitions to Section

A Markov growth process for Macdonald's distribution on reduced words

We give an algorithmic-bijective proof of Macdonald's reduced word identity in the theory of Schubert polynomials, in the special case where the permutation is dominant. Our bijection uses a novel application of David Little's generalized bumping algorithm. We also describe a Markov growth process for an associated probability distribution on reduced words. Our growth process can be implemented efficiently on a computer and allows for fast sampling of reduced words. We also discuss various partial generalizations and links to Little's work on the RSK algorithm.Comment: 16 pages, 5 figure

Perceiving Smellscapes

We perceive smells as perduring complex entities within a distal array that might be conceived of as smellscapes. However, the philosophical orthodoxy of Odor Theories has been to deny that smells are perceived as having a distal location. Recent challenges have been mounted to Odor Theories’ veracity in handling the timescale of olfactory perception, how it individuates odors as a distal entities, and their claim that olfactory perception is not spatial. The paper does not aim to dispute these criticisms. Rather, what will be shown is that Molecular Structure Theory, a refinement of Odor Theory, can be further developed to handle these challenges. The theory is further refined by focusing on distal perception that requires considering the perceptual object as mereologically complex persisting odor against a background scene conceived of as a smellscape. What will be offered is an expansion of Molecular Structure Theory to account for distal smell perception within natural environments

Smell's puzzling discrepancy: Gifted discrimination, yet pitiful identification

Mind &Language, Volume 35, Issue 1, Page 90-114, February 2020

ESTIMATION OF A DIRECT UTILITY FUNCTION FOR FOOD EXPENDITURES

Food Consumption/Nutrition/Food Safety,