Supersymmetry and Combinatorics

We show how a recently proposed supersymmetric quantum mechanics model leads to non-trivial results/conjectures on the combinatorics of binary necklaces and linear-feedback shift-registers. Fermi statistics plays a crucial role by projecting out certain states/necklaces by virtue of Pauli's famous exclusion principle. Some of our results can be rephrased in terms of generalizations of the well-known Witten index. Binary necklaces (BNLs) and linear-feedback shift-registers (LFSRs) are much studied objects in the branch of mathematics known as combinatorics (for standard textbooks, see for example[1], [2]). Let us start by recalling what is known in the literature about counting and/or enumerating these objects. For BNLs the most relevant results are from Polya's theory of counting [1]. For reasons related to the physical model described below, we shall denote by B the number of beads of one type and by F the number of beads of the second type in the BNL. The total number of beads will be denoted by n (n = B + F). The number of BNLs with some given B and F is given by Polya's formula: (1) where d|B, F means that d divides B and F , and ϕ(d) is Euler's " totient " function, counting the numbers in 1, 2, ..., d − 1 relatively prime to d. After summing eq. (1) over B while keeping n fixed, we obtain the well-known MacMahon's formula [1]: (2) N BNL (n) = B+F =n N BNL (B, F) = 1 n d|n ϕ(d) 2 n/d. The numbers generated by eq. (2) define a series of integers known [3] as A000031(n). Its definition in [3] is indeed: A000031(n) = Number of n-bead necklaces with 2 colours when turning over is not allowed, meaning that one distinguishes BNLs that differ by reversal of the ordering of the beads. Other interesting integer series are provided by various kinds of LFSRs. In particular, the expression for series A000016(n + 1) defined as [3]: A000016(n+1) = Number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the sum of its contents. Twice that number bears an amusing similarity to eq. (2): (3) 2A000016(n) ≡ 2N LFSR (n) = 1 n d|n d odd ϕ(d) 2 n/d , and is also known to coincide with A063776(n): A063776(n) = Number of subsets of {1, 2,. .. , n} which sum to 0 mod n. In this short note …