Generating All Partitions: A Comparison Of Two Encodings

Abstract

Integer partitions may be encoded as either ascending or descending compositions for the purposes of systematic generation. Many algorithms exist to generate all descending compositions, yet none have previously been published to generate all ascending compositions. We develop three new algorithms to generate all ascending compositions and compare these with descending composition generators from the literature. We analyse the new algorithms and provide new and more precise analyses for the descending composition generators. In each case, the ascending composition generation algorithm is substantially more efficient than its descending composition counterpart. We develop a new formula for the partition function p(n) as part of our analysis of the lexicographic succession rule for ascending compositions.

Parametrizing Arf numerical semigroups

“…Let c be a positive integer. In light of Corollary 2 and Proposition 3, in order to calculate the set of Arf numerical semigroups with conductor c we only have to calculate some specific integer partitions of c, and then their images via the map S. We can compute the set of integer partitions of c with the help of [14] or the built-in GAP command partitions, and either filter those having 1's or while constructing them avoid 1's in the partition. However, the number of partitions grows exponentially (for instance NrPartitions(100) in GAP yields 190569292), and then we must choose which partitions are Arf sequences.…”

“…-If r = w(2), then w(2) < w(4) and thus w(4) = w(2) + 2 by (14) If c is an integer such that c ≥ 6 and c ≡ 1 (mod 6)), then Proposition 22 can be used to count Arf numerical semigroups with multiplicity 6 and conductor c. if c ≡ 5 (mod 6).…”

“…we have w(5) = c − k + 5 and (13)w(1) = c + 1 if k = 0, c − k + 7 if k = 0.Also, w(4) ≤ w(2) + 2 and w(2) = w(6 − 4) ≤ w(6 − 2) + (6 − 2) = w(4) + 4 by Lemma 15. Combining these two inequalities we get(14) w(4) − 2 ≤ w(2) ≤ w(4) + 4.Let us also note that w(i) ≤ c + 5 for all i ∈ {1, 2, 3, 4, 5}. (i) If c ≡ 0 (mod 6), then w(1) = c + 1 and w(5) = c + 5 by(13).…”

Programming the Partition Method for a Power Series Expansion

“…Kelleher and O'Sullivan in Refs. [14] and [15] have developed algorithms for generating partitions in ascending order, but not in lexicographic order, one of which has been created as the C/C++-code appearing above. When this code is run for the partitions summing to 5, the following output is obtained:…”

Fast Algorithm for Generating Ascending Compositions

“…The choice of the way in which the partitions are represented is crucial for the efficiency of their generating algorithm. Kelleher [4,5] approaches the problem of generating partitions through ascending compositions and proves that the algorithm AccelAsc is more efficient than any other previously known algorithms for generating integer partitions.…”