An analysis of two related loop-free algorithms for generating integer partitions

Fenner, Trevor; Loizou, G.

doi:10.1007/bf00261261

Cited by 11 publications

(3 citation statements)

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“…Algorithms to generate descending compositions in lexicographic order have also been published. See Knuth [25, p.147] and Zoghbi & Stojmenović [62] for implementations using the sequence representation; Reingold, Nievergelt & Deo [42, p.193] and Fenner & Loizou [16] for implementations using the multiplicity representation; and Klimko [24] for an implementation using the part-count representation. Fenner & Loizou's tree construction operations [15] can be used to generate descending compositions in several other orders.…”

Section: Related Workmentioning

confidence: 99%

Generating All Partitions: A Comparison Of Two Encodings

Kelleher¹,

O’Sullivan²

2009

Preprint

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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.

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Section: Related Workmentioning

confidence: 99%

Generating All Partitions: A Comparison Of Two Encodings

Kelleher¹,

O’Sullivan²

2009

Preprint

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“…Algorithms for generating integer partitions in standard orders such as lexicographic and antilexicographic were presented in [FL80] and [NW78]. The performance of the algorithms in [FL80] is analyzed in [FL81].…”

Section: Integer Partitionsmentioning

confidence: 99%

A Survey of Combinatorial Gray Codes

Savage¹

1997

SIAM Rev.

387

254

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The term combinatorial Gray code was introduced in 1980 to refer to any method for generating combinatorial objects so that successive objects differ in some pre-specified, small way. This notion generalizes the classical binary reflected Gray code scheme for listing n-bit binary numbers so that successive numbers differ in exactly one bit position, as well as work in the 1960's and 70's on minimal change listings for other combinatorial families, including permutations and combinations.The area of combinatorial Gray codes was popularized by Herbert Wilf in his invited address at the SIAM Discrete Mathematics Conference in 1988 and his subsequent SIAM monograph in which he posed some open problems and variations on the theme. This resulted in much recent activity in the area and most of the problems posed by Wilf are now solved.In this paper, we survey the area of combinatorial Gray codes, describe recent results, variations, and trends, and highlight some open problems.

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“…Many algorithms for the enumeration or random generation of combinatorial objects can be found in the literature. In particular, sequential algorithms for the generation of all partitions of a given integer, either in general or subject to some restrictions, may be found in [11,15,12,[4][5][6]17]. As mentioned above, Ref.…”

Section: Introductionmentioning

confidence: 99%