A solution to one of Knuth's permutation problems

TL;DR

This paper solves a permutation problem posed by Knuth, analyzing the volume distribution of dissected n-dimensional boxes and revealing a connection to Catalan numbers, with implications for geometric combinatorics.

Contribution

It provides a complete characterization of volume equivalence classes in the dissection of n-dimensional boxes along coordinate planes, linking to Catalan numbers.

Findings

Number of distinct volumes equals the nth Catalan number.

Volumes are grouped into C_n classes with equal measure.

The dissection pattern is explicitly described.

Abstract

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.