Wilf’s conjecture in fixed multiplicity

Abstract

We give an algorithm to determine whether Wilf's conjecture holds for all numerical semigroups with a given multiplicity m, and use it to prove Wilf's conjecture holds whenever m ≤ 18. Our algorithm utilizes techniques from polyhedral geometry, and includes a parallelizable algorithm for enumerating the faces of any polyhedral cone up to orbits of an automorphism group. We also introduce a new method of verifying Wilf's conjecture via a combinatorially-flavored game played on the elements of a certain finite poset. arXiv:1903.04342v2 [math.CO]

A Graph-Theoretic Approach to Wilf's Conjecture

“…Now if u were not primitive, say if u = u 1 u 2 with u 1 , u 2 ∈ V , then u 3 1 u 3 2 ∈ X, and this would yield at least two distinct edges with same weight, e.g. {u 1 , u 2 1 u 3 2 } and {u 2 1 , u 1 u 3 2 }. Hence u ∈ V ∩ P , as claimed.…”

“…Since |V ∩ D| assumption, it follows from Proposition 54 that V ∩ D consists of leaves, each of the form x 2 with x ∈ V ∩ P as unique neighbor. Therefore V ∩ D = {y 1 , y 2 }, and since both have x 1 as unique neighbor, this implies y 1 = y 2 = x 2 1 , an absurdity since y 1 , y 2 are distinct. Hence the present case, namely n = 5, k = 4, |X ∩ D| 8, |V ∩ D| 2 and λ = 3, cannot occur.…”

“…Remark 68. Corollary 67 has just been improved with a verification of Wilf 's conjecture up to multiplicity m 18, by computer calculations with a specially developed algorithm based on the Kunz polytope and polyhedral geometry [2].…”

“…Wilf's conjecture, as it is now known, has been verified in several cases, including when |P | 3, or c 3m, or m 18, or |L| 12, or |P | m/2. See Delgado [5] for an extensive recent survey of partial results on Wilf's conjecture, and [1,2,4,9,11,13,14,15,17,20,21,22] for some relevant papers. The verification in case |P | m/2 is due to Sammartano [20] in 2012.…”

“…Thus z 1 wv ∈ X ∩ D, implying {z 1 w, z 1 v, wv} ⊆ V ∩ D. Therefore z 1 w = z 1 v = wv, whence z 1 = w = v and u = z 21 . Moreover z 1 ∈ P , for if z 1 had proper factors in V , this would imply z 1 ∈ V ∩ D, contradicting the equality V ∩ D = {z2 1 }.Proposition 53. We have |X ∩ D| |E(G)| − deg(u) for all u ∈ V ∩ D such that u = x 2 with x ∈ P .…”