Compositions into Powers of b: Asymptotic Enumeration and Parameters

Krenn, Daniel; Wagner, Stephan

doi:10.1007/s00453-015-0061-3

Cited by 5 publications

(4 citation statements)

References 17 publications

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“…If * is the arithmetic, geometric, or harmonic mean, then s a n ( * ) = C n−1 for all n ≥ 1 (see Csákány, Waldhauser [5]). However, as we are going to show next, its ac-spectrum agrees with an interesting sequence in OEIS [19, A007178], which enumerates different ways to write 1 as an ordered sum of n powers of 2 (i.e., compositions of 1 into powers of 2) and is also related to the so-called prefix codes or Huffman codes (see, e.g., Even and Lempel [6], Giorgilli and Molteni [11], Knuth [12], Krenn and Wagner [13] and Lehr, Shallit and Tromp [14]). Proposition 4.4.…”

Section: 2supporting

confidence: 53%

The associative-commutative spectrum of a binary operation

Jia¹,

Lehtonen²

2022

Preprint

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We initiate the study of a quantitative measure for the failure of a binary operation to be commutative and associative. We call this measure the associative-commutative spectrum as it extends the so-called associative spectrum (also known as the subassociativity type), which measures the nonassociativity of a binary operation. In fact, the associative-commutative spectrum (resp. associative spectrum) is the cardinality of the symmetric (resp. nonsymmetric) operad obtained naturally from a groupoid (a set with a binary operation). In this paper we provide some general results on the associative-commutative spectrum, precisely determine this measure for certain binary operations, and propose some problems for future study.

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Section: 2supporting

confidence: 53%

The associative-commutative spectrum of a binary operation

Jia¹,

Lehtonen²

2022

Preprint

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“…Krenn and Wagner [19] showed that the number of full-support dyadic distributions on X n is asymptotic to αγ n−1 n!, where α ≈ 0.296 and γ ≈ 1.193, implying that the number of dyadic distributions on X n is asymptotic to αe 1/γ γ n−1 n!. Boyd [4] showed that the number of monotone full-support dyadic distributions on X n is asymptotic to βλ n , where β ≈ 0.142 and λ ≈ 1.794, implying that the number of monotone dyadic distributions on X n is asymptotic to β(1 + λ) n .…”

Section: Reduction To Maximum Relative Densitymentioning

confidence: 99%

Twenty (simple) questions

Dagan

Filmus

Gabizon³

et al. 2017

Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing

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A basic combinatorial interpretation of Shannon's entropy function is via the "20 questions" game. This cooperative game is played by two players, Alice and Bob: Alice picks a distribution π over the numbers {1, . . . , n}, and announces it to Bob. She then chooses a number x according to π, and Bob attempts to identify x using as few Yes/No queries as possible, on average. An optimal strategy for the "20 questions" game is given by a Huffman code for π: Bob's questions reveal the codeword for x bit by bit. This strategy finds x using fewer than H(π) + 1 questions on average. However, the questions asked by Bob could be arbitrary. In this paper, we investigate the following question: Are there restricted sets of questions that match the performance of Huffman codes, either exactly or approximately?Our first main result shows that for every distribution π, Bob has a strategy that uses only questions of the form "x < c?" and "x = c?", and uncovers x using at most H(π) + 1 questions on average, matching the performance of Huffman codes in this sense. We also give a natural set of O(rn 1/r ) questions that achieve a performance of at most H(π) + r, and show that Ω(rn 1/r ) questions are required to achieve such a guarantee.Our second main result gives a set Q of 1.25 n+o(n) questions such that for every distribution π, Bob can implement an optimal strategy for π using only questions from Q. We also show that 1.25 n−o(n) questions are needed, for infinitely many n. If we allow a small slack of r over the optimal strategy, then roughly (rn) Θ(1/r) questions are necessary and sufficient.We summarize this with the following meta-question, which guides this work: Are there "nice" sets of queries Q such that for any distribution, there is a "high quality" strategy that uses only queries from Q?Formalizing this question depends on how "nice" and "high quality" are quantified. We consider two different benchmarks for sets of queries:1. An information-theoretical benchmark: A set of queries Q has redundancy r if for every distribution π there is a strategy using only queries from Q that finds x with at most H(π)+r queries on average when x is drawn according to π. A combinatorial benchmark:A set of queries Q is r-optimal (or has prolixity r) if for every distribution π there is a strategy using queries from Q that finds x with at most Opt(π) + r queries on average when x is drawn according to π, where Opt(π) is the expected number of queries asked by an optimal strategy for π (e.g. a Huffman tree).Given a certain redundancy or prolixity, we will be interested in sets of questions achieving that performance that (i) are as small as possible, and (ii) allow efficient construction of high quality strategies which achieve the target performance. In some cases we will settle for only one of these properties, and leave the other as an open question.Information-theoretical benchmark. Let π be a distribution over X. A basic result in information theory is that every algorithm that reveals an unknown element x drawn according to π (in...

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“…A. Sellers [76] and D. Krenn and S. Wagner [97] (which deals mostly with m-ary compositions). It is easy to see that the number f bp In conclusion of Section 3 and of our article we turn to cancellative problems related to the initial Example 5.…”

Section: Proofmentioning

confidence: 99%

“…A. Sellers [76] and D. Krenn and S. Wagner [97] (which deals mostly with m-ary compositions). It is easy to see that the number f bp (n) := f mp (n, 2) of binary partitions of n follows the recurrence f bp (0) = 1 and f bp (n) = f bp (n − 1) + f bp (n/2) for n ≥ 1 (where f bp (n/2) = 0 if n/2 ∈ N 0 ).…”

mentioning

confidence: 99%

Some general results in combinatorial enumeration

Klazar

2010

Permutation Patterns

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For enumerative problems, i.e. computable functions f : N → Z, we define the notion of an effective (or closed) formula. It is an algorithm computing f (n) in the number of steps that is polynomial in the combined size of the input n and the output f (n), both written in binary notation. We discuss many examples of enumerative problems for which such closed formulas are, or are not, known. These problems include (i) linear recurrence sequences and holonomic sequences, (ii) integer partitions, (iii) pattern-avoiding permutations, (iv) triangle-free graphs and (v) regular graphs. In part I we discuss problems (i) and (ii) and defer (iii)-(v) to part II. Besides other results, we prove here that every linear recurrence sequence of integers has an effective formula in our sense. Definition 1.1 (PIO formula). For a counting function f : N → Z, a PIO formula is an algorithm, called a PIO algorithm, that for some constants c, d ∈ N for every input n ∈ N computes the output f (n) ∈ Z in at most c · m(n) d = O(m(n) d ) = poly(m(n)) steps, where m(n) = m f (n) := log(1 + n) + log(2 + |f (n)|) measures the combined complexity of the input and the output. Similarly for counting function f : X → Z defined on a subset X ⊂ N.We think this is the precise and definitive notion of a "closed formula", and the yardstick one should use, possibly with some ramifications or weakenings, to determine if a solution to an enumerative problem is effective. We call counting functions possessing PIO formulas shortly PIO functions. Definition 1.1 repeats

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Compositions into Powers of b: Asymptotic Enumeration and Parameters

Cited by 5 publications

References 17 publications

The associative-commutative spectrum of a binary operation

The associative-commutative spectrum of a binary operation

Twenty (simple) questions

Some general results in combinatorial enumeration

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