Abstract
The smallest part is a rational function. This result is similar to the closely related case of partitions with fixed differences between largest and smallest parts which has recently been studied through analytic methods by Andrews, Beck, and Robbins. Our approach is geometric: We model partitions with bounded differences as lattice points in an infinite union of polyhedral cones. Surprisingly, this infinite union tiles a single simplicial cone. This construction then leads to a bijection that can be interpreted on a purely combinatorial level.
Highlights
A partition of a non-negative integer n is a weakly non-increasing finite sequence of positive whole numbers λ1 λ2 · · · λk > 0 such that n = λ1 + λ2 + · · · + λk.The integers λ1, λ2, · · ·, λk are called the parts of the partition
Our strategy is this: Just as in the case t = 0, we identify each set in the infinite union (2) as the set of lattice points in a half-open polyhedral cone
Theorem 3 implies this arithmetic corollary, but it leads to a bijective proof of Theorem 1: We can interpret (1) as counting partitions with bounded differences and (3) as counting pairs (λ, ) where λ is a non-empty partition with largest part at most t and is a non-negative multiple of t
Summary
We develop a polyhedral model of partitions with bounded differences that allows us to interpret the identity of (1) and (3) in terms of a tiling of a polyhedral cone in Theorem 3 This geometric result immediately implies Theorem 1 using different methods than the q-series manipulations employed in [3]. In contrast to the case t = 0, these cones Cm are 2-dimensional and they tile the half-open quadrant R>0 × R 0, which is itself a single half-open simplicial cone This immediately allows us to read off the desired rational function expression (3). We have observed that the infinite union of cones (4) is a single simplicial cone (5) This allows us to read off the generating function immediately, namely xv =.
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