On cardinalities of $k$-abelian equivalence classes

TL;DR

This paper investigates the sizes of $k$-abelian equivalence classes of words, establishing bounds on the number of singleton classes and connecting these to cycle decompositions of de Bruijn graphs, with implications for combinatorics on words.

Contribution

It provides bounds on the number of singleton $k$-abelian classes and links these to cycle decompositions of de Bruijn graphs, proposing conjectures on the bounds' sharpness.

Findings

Number of singleton classes grows as O(n^{N_m(k-1)-1})

Connection established between singleton classes and de Bruijn graph cycles

Verification of Gray code conjecture for necklaces up to length 15

Abstract

Two words $u$ and $v$ are $k$-abelian equivalent if, for each word $x$ of length at most $k$, $x$ occurs equally many times as a factor in both $u$ and $v$. The notion of $k$-abelian equivalence is an intermediate notion between the abelian equivalence and the equality of words. In this paper, we study the equivalence classes induced by the $k$-abelian equivalence, mainly focusing on the cardinalities of the classes. In particular, we are interested in the number of singleton $k$-abelian classes, i.e., classes containing only one element. We find a connection between the singleton classes and cycle decompositions of the de Bruijn graph. We show that the number of classes of words of length $n$ containing one single element is of order $\mathcal O(n^{N_m(k-1)-1})$, where $N_m(l) = \tfrac{1}{l}\sum_{d\mid l} \varphi(d)m^{l/d}$ is the number of necklaces of length $l$ over an $m$-ary alphabet. We conjecture that the upper bound is sharp. We also remark that, for $k$ even and $m = 2$, the lower bound $\Omega(n^{N_m(k-1)-1})$ follows from an old conjecture on the existence of Gray codes for necklaces of odd length. We verify this conjecture for necklaces of length up to 15.