Cardinality of Balls in Permutation Spaces

TL;DR

This paper investigates the growth of the number of permutations within a certain distance in permutation spaces, providing conditions for polynomial growth and explicit formulas for specific distances.

Contribution

It offers a sufficient condition for polynomial growth of permutation balls and derives explicit formulas for sphere cardinalities under the  distance.

Findings

Cardinality of permutation balls can grow polynomially under certain conditions.

Explicit polynomial formulas for sphere sizes in the  distance are derived.

High degree terms of these polynomials are determined.

Abstract

For a right invariant distance on a permutation space $S_n$ we give a sufficient condition for the cardinality of a ball of radius $R$ to grow polynomially in $n$ for fixed $R$. For the distance $\ell_1$ we show that for an integer $k$ the cardinality of a sphere of radius $2k$ in $S_n$ (for $n \geqslant k$) is a polynomial of degree $k$ in $n$ and determine the high degree terms of this polynomial.