The largest missing value in a composition of an integer

Archibald, Margaret; Knopfmacher, Arnold

doi:10.1016/j.disc.2011.01.012

Cited by 7 publications

(8 citation statements)

References 10 publications

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“…By substituting p = 1/2 in Theorem 3.1, we can check this against the corresponding result in [2]. That result was derived by simpler means, by using the explicit expression for C(z) known for the case p = 1/2.…”

Section: The Case P = 1/2mentioning

confidence: 89%

“…Previously, these questions have been studied in the special case of p = 1/2 in [2], and subsequently the p = 1/2 case was revisited in [12]. However, neither of the methods of [2] or [12] apply to the more general case of 0 < p < 1 as studied in the present paper.…”

Section: Introductionmentioning

confidence: 88%

“…Remark. In the simpler case of p = 1/2, the tiny additional term [δ E ] 2 0 was computed explicitly in [2] and was shown to be of magnitude 10 −12 . By including this known tiny term from the p = 1 2 case, we have an identity (proved in the subsection below) to show that this result corresponds to the variance in [2].…”

Section: Largest Missing Value: Variancementioning

confidence: 99%

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The Largest Missing Value in a Sample of Geometric Random Variables

Archibald

Knopfmacher

2014

Combinator. Probab. Comp.

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We consider samples of n geometric random variables with parameter 0 < p < 1, and study the largest missing value, that is, the highest value of such a random variable, less than the maximum, that does not appear in the sample. Asymptotic expressions for the mean and variance for this quantity are presented. We also consider samples with the property that the largest missing value and the largest value which does appear differ by exactly one, and call this the LMV property. We find the probability that a sample of n variables has the LMV property, as well as the mean for the average largest value in samples with this property. The simpler special case of p = 1/2 has previously been studied, and verifying that the results of the present paper coincide with those previously found for p = 1/2 leads to some interesting identities.

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Section: The Case P = 1/2mentioning

confidence: 89%

Section: Introductionmentioning

confidence: 88%

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The Largest Missing Value in a Sample of Geometric Random Variables

Archibald

Knopfmacher

2014

Combinator. Probab. Comp.

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“…is a uniform random composition of n. Using this fact, Hitczenko and others were able to determine the asymptotic distributions of a variety of random variables defined on the space of compositions of n with a uniform probability measure [1,15,16,17,18,19,20].…”

Section: Introductionmentioning

confidence: 99%

Part-products of S-restricted integer compositions

Schmutz

Shapcott

2013

APPL ANAL DISCRETE M

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If S is a cofinite set of positive integers, an "S-restricted composition of n" is a sequence of elements of S, denoted λ = (λ1, λ2, . . . ), whose sum is n. For uniform random S-restricted compositions, the random variable B( λ) = i λi is asymptotically lognormal. (A precise statement of the theorem includes an error term to bound the rate of convergence.) The proof is based upon a combinatorial technique for decomposing a composition into a sequence of smaller compositions.2010 Mathematics Subject Classification. 05A16, 60C05, 60F05.

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“…Suppose we require that the inequalities be strict. This can be done by simply changing 0 < a ≤ b to 0 < a < b in (1). Now, however, we can include all strict alternating compositions in one φ instead of using C {φ,φ ′ } .…”

Section: Introductionmentioning

confidence: 99%

Locally Restricted Compositions IV. Nearly Free Large Parts and Gap-Freeness

Bender

Canfield²,

Gao

2012

Electron. J. Combin.

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We define the notion of asymptotically free for locally restricted compositions, which means roughly that large parts can often be replaced by any larger parts. Two well-known examples are Carlitz and alternating compositions. We show that large parts have asymptotically geometric distributions. This leads to asymptotically independent Poisson variables for numbers of various large parts. Based on this we obtain asymptotic formulas for the probability of being gap free and for the expected values of the largest part, number of distinct parts and number of parts of multiplicity k, all accurate to o(1).Dedicated to the memory of Herb Wilf.

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The largest missing value in a composition of an integer

Cited by 7 publications

References 10 publications

The Largest Missing Value in a Sample of Geometric Random Variables

The Largest Missing Value in a Sample of Geometric Random Variables

Part-products of S-restricted integer compositions

Locally Restricted Compositions IV. Nearly Free Large Parts and Gap-Freeness

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