Partitions in the image of preₖ: complete searches, computed values, and the classification of preₖ(n)=1

Abstract

Ballantine, Beck and Merca attach to a partition λ and an integer k ≥ 2 the partition pk(k)(λ) whose parts are the products of the parts of λ over all k-element sets of indices; its weight is the elementary symmetric function e(k)(λ). Following Devnani and Eyyunni, pk(k)(n) denotes the number of partitions of n that lie in the image of pk(k). Garg, and independently Thomas and Tung, proved that pk(2)(n)=1 exactly for n ∈ {1,2,4}; no value of pk(k)(n) appears in print for any k ≥ 3, and the analogue of that theorem at k ≥ 3 was not known. We prove that the natural pruned depth-first search for the partitions λ with e(k)(λ)=n is complete for k=2,3,4,5, so that pk(k)(n) is an effectively computable function of n rather than a count over a truncated search, and we tabulate pk(2)(n) for n ≤ 460 with further values up to 1000, pk(3)(n) for n ≤ 400, and pk(5)(n) for n ≤ 500 with further values up to 1000. Our main result is the exact analogue of Garg's theorem at k=3: pk(3)(n)=1 for exactly 28 integers n, the largest being 321, at every n ≥ 1 and with no range restriction. The proof removes pk(3) from the problem and covers the complement of the exceptional set by 328 arithmetic progressions. At k=4 and k=5 we determine the exceptional set on [1,10⁵] and on [1,2 · 10⁴] (respectively 171 integers with largest 5942, and 1103 integers with largest 19969), in each case with the half asserting pk(k)(n)=1 holding at every n; an exhaustive computation shows that no covering system exists at k=4 with modulus lattice of least common multiple at most 3 · 10⁵, and partial coverings instead confine {n:pk(4)(n)=1} to at most 11.63 modulo 240240 and {n:pk(5)(n)=1} to at most 43.12 machine-checked in Lean 4.

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