Cayley Compositions, Partitions, Polytopes, and Geometric Bijections

In 1857, Cayley showed that certain sequences, now called Cayley compositions, are equinumerous with certain partitions into powers of 2. In this paper we give a simple bijective proof of this result and a geometric generalization to equality of Ehrhart polynomials between two convex polytopes. We then apply our results to give a new proof of Braun’s conjecture proved recently by the authors [KP2]. Introduction and main results Partition Theory is a classical field with a number of advanced modern results and applications. Its long and tumultuous history left behind a number of beautiful results which are occasionally brought to light to wide acclaim. The story of the so called Cayley compositions is a prime example of this. Introduced and studied by Cayley in 1857 [Cay], they were rediscovered by Minc [Minc], and remained largely forgotten until Andrews, Paule, Riese and Strehl [APRS] resurrected and christened them in 2001. This is when things became really interesting. Theorem 1 (Cayley, 1857). The number of integer sequences (a1, . . . , an) such that 1 ≤ a1 ≤ 2, and 1 ≤ ai+1 ≤ 2ai for 1 ≤ i < n, is equal to the total number of partitions of integers N ∈ {0, 1, . . . , 2 − 1} into parts 1, 2, 4, . . . , 2n−1. Our first result is a long elusive bijective proof of Cayley’s theorem, and its several extensions. Our bijection construction is geometric, based on our approach in [P1]. Denote by An the set of sequences (a1, . . . , an) satisfying the conditions of the theorem, which are called Cayley compositions. Denote by Bn the set of partitions into powers of 2 as in the theorem, which we call Cayley partitions. Now Theorem 1 states that |An| = |Bn|. For example, A2 = { (1, 1), (1, 2), (2, 1), (2, 2), (2, 3), (2, 4) } , B2 = { 21, 2, 1, 1, 1, ∅ } , so |A2| = |B2| = 6. Following [BBL], define the Cayley polytope An to be the convex hull of all Cayley compositions (a1, . . . , an) ∈ R. The main result of this paper is the following geometric extension of Theorem 1. Recall that the Ehrhart polynomial EP (t) of a lattice polytope P ⊂ R is defined by EP (k) = # {k P ∩ Z} , Date: November 20, 2013. Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia. †Department of Mathematics, UCLA, Los Angeles, CA 90095, USA, Email: [email protected] Phone: +1-310-825-4701, Fax: +1-310-206-6673. 1 2 MATJAŽ KONVALINKA AND IGOR PAK where kP denotes the k-fold dilation of P , k ∈ N (see e.g. [Bar]). Theorem 2. Let Bn be the set of Cayley partitions, where a partition of the form (2n−1)m1(2n−2)m2 . . . 1n is identified with an integer point (m1,m2, . . . ,mn) ∈ R. Now let Bn = convBn. Then EAn(t) = EBn(t). In particular, when t = 1, we obtain Cayley’s theorem. Our proof is based on an explicit volume-preserving map φ : Bn → An, which satisfies a number of interesting properties. In particular, when restricted to integer points, this map gives the bijection φ : Bn → An mentioned above (see Proposition 6). In [BBL], Ben Braun made an interesting conjecture about the volume of An, which was recently proved by the authors [KP2]. Denote by Cn the set of connected graphs on n nodes, and let Cn = ∣∣Cn∣∣. Theorem 3 ([KP2], formerly Braun’s conjecture). Let An ⊂ R be the set of Cayley compositions, and let An = convAn be the Cayley polytope. Then volAn = Cn+1/n!. Combined with Theorem 2, we immediately have volBn = volAn, and conclude: Corollary 4. Let Bn be the polytope defined above. Then volBn = Cn+1/n!. Curiously, one can also use volBn = volAn in reverse, and derive Theorem 3 from Theorem 2 and known results on Stanley-Pitman polytopes (see below). The rest of this paper is structured as follows. In Section 1 we prove Theorems 1 and 2 using an explicit bijection φ. Some applications are given in Section 2, followed by a new proof of Theorem 3 in Section 3. We finish with final remarks in Section 4. 1. Bijection construction Recall from [BBL, KP2] (or observe directly from the definition) that Cayley polytope An ⊂ R is defined by the following inequalities: 1 ≤ x1 ≤ 2, 1 ≤ x2 ≤ 2x1 , . . . , 1 ≤ xn ≤ 2xn−1 .