Necklaces, subset sums, and cyclic permutations

Abstract

It is a well known that, for odd $n$, the number of subsets of $\{1,2,\dots,n\}$ the sum of whose elements is divisible by $n$ equals the number of binary necklaces of length $n$. In this paper generalize this result in two directions. On the one hand, we introduce a parameter $r$ so that requiring the subset sums to be congruent to $r$ modulo $n$ translates into imposing some periodicity conditions on the necklaces. On the other hand, we refine these relations by the size $k$ of the subset, showing that it matches the number of ones in the necklace. We describe the precise conditions on $n$, $k$ and $r$ for which the equalities hold. The classical results correspond to the case $r=0$. When $r=1$, our identity is related to a conjecture of Baker et al. connecting subsets the sum of whose elements is congruent to $1$ modulo $n$ and unimodal permutations which consist of one cycle. We prove this conjecture using generating functions. Finding bijective proofs of most of our identities remains an open problem.

Figures (1)

• Figure 1: A diagram of the relationships described in Section \ref{['sec:r=1']}, for $1\le k\le n-1$.

Theorems & Definitions (36)

• Conjecture 1.1: Baker
• Definition 1
• Lemma 2.1
• proof
• Definition 2
• Definition 3
• Theorem 3.1
• Lemma 3.2: Thibon
• Lemma 3.3
• proof