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Abstract
Let K_n denote the smaller mode of the nth row of Stirling numbers of the second kind S(n, k). Using a probablistic argument, it is shown that for all n>=2, [exp(w(n))]-2<=K_n<=[exp(w(n))]+1, where [x] denotes the integer part of x, and w(n) is Lambert's W-function. Comment: accepted by Discrete Mathematics
... More recently, non-asymptotic bounds for k n have been obtained. In this regard, Yu [14] gave ...
... The difficulty in proving this inequality stems from the fact that both sides in (14) are positive and, apparently, have the same order of magnitude. This implies that a very precise estimate of Q(n, ν n ) is needed in order to prove (or disprove) Wegner's conjecture. ...
The Stirling numbers of the second kind S(n, k) satisfy S(n, 0) < · · · < S(n, kn) ≥ S(n, kn + 1) > · · · > S(n, n). A long standing conjecture asserts that there exists no n ≥ 3 such that S(n, kn) = S(n, kn + 1). In this note, we give a characterization of this conjecture in terms of multinomial probabilities, as well as sufficient conditions on n ensuring that S(n, kn) > S(n, kn + 1). Mathematics Subject Classification. 05A19, 11B73, 60E05. Keywords. Stirling number of the second kind, uniqueness conjecture, multinomial law.
... More recently, non-asymptotic bounds for k n have been obtained. In this regard, Yu [14] gave ...
... The difficulty in proving this inequality stems from the fact that both sides in (14) are positive and, apparently, have the same order of magnitude. This implies that a very precise estimate of Q(n, ν n ) is needed in order to prove (or disprove) Wegner's conjecture. ...
The Stirling numbers of the second kind S(n, k) satisfy S(n,0)<⋯<S(n,kn)≥S(n,kn+1)>⋯>S(n,n).A long standing conjecture asserts that there exists no n≥3 such that S(n,kn)=S(n,kn+1). In this note, we give a characterization of this conjecture in terms of multinomial probabilities, as well as sufficient conditions on n ensuring that S(n,kn)>S(n,kn+1).
... Moreover, if a polynomial n i=0 u i x i with nonnegative coefficients has only real zeros, then the sequence u i , 0 ≤ i ≤ n, is log-concave with no internal zeros. Unimodal, log-concave and logconvex sequences arise naturally in many problems in combinatorics and elsewhere; see [4, 9] and [13]–[22], for example. Unimodality properties of sequences associated with Pascal's triangle have always been of interest ([17, 18]). ...
Two conjectures of Su and Wang (2008) concerning binomial coefficients are proved. For n⩾k⩾0 and b>a>0, we show that the finite sequence Cj=(n+jak+jb) is a Pólya frequency sequence. For n⩾k⩾0 and a>b>0, we show that there exists an integer m⩾0 such that the infinite sequence (n+jak+jb), j=0,1,… , is log-concave for 0⩽j⩽m and log-convex for j⩾m. The proof of the first result exploits the connection between total positivity and planar networks, while that of the second uses a variation-diminishing property of the Laplace transform.
Here we study the distribution of randomly generated partitions of the set of amino acid encoding codons. Some results are an application from a previous work, about the Stirling numbers of the second kind and triplet codes, both to the cases of triplet codes having four stop codons, as in mammalian mitochondrial genetic code, and hypothetical doublet codes.
Extending previous results, in this work it is found that the most probable number of blocks of synonymous codons, in a genetic code, is similar to the number of amino acids when there are four stop codons, as well as it could be for a primigenious doublet code. Also it is studied the integer partitions associated to patterns of synonymous codons and it is shown, for the canonical code, that the standard deviation inside an integer partition is one of the most probable.
We think that, in some early epoch, the genetic code might have had a maximum of the disorder or entropy, independent of the assignment between codons and amino acids, reaching a state similar to “code freeze” proposed by Francis Crick. In later stages, maybe deterministic rules have reassigned codons to amino acids, forming the natural codes, such as the canonical code, but keeping the numerical features describing the set partitions and the integer partitions, like a “fossil numbers”; both kinds of partitions about the set of amino acid encoding codons.
Given a genetic code formed by 64 codons, we calculate the number of partitions of the set of encoding amino acid codons. When there are 0-3 stop codons, the results indicate that the most probable number of partitions is 19 and/or 20. Then, assuming that in the early evolution the genetic code could have had random variations, we suggest that the most probable number of partitions of the set of encoding amino acid codons determined the actual number 20 of standard amino acids.
The Stirling numbers of the second kind are asymptotically normal. This result is similar to results achieved by Feller [1] and Goncarov [2] for other combinatorial distributions. Here the technique of proof is different; one of the most general forms of the central limit theorem is used. Interesting qualitative information about the Stirling numbers is also obtained from this result. Asymptotic estimates on the value of are given.
Say an integer n is exceptional if the maximum Stirling number of the second kind S(n,k) occurs for two (of necessity consecutive) values of k. We prove that the number of exceptional integers less than or equal to x is O(x 1/2+ε ), for any ε>0. We derive a similar result for partitions of n into exactly k integers.
Let S(n, k) denote Stirling numbers of the second kind, and Kn be the integer(s) such that S(n, Kn) ⩾ S(n, k) for all k. We determine the value(s) of Kn to within a maximum error of 1.
Say that an integer n is exceptional if the maximum Stirling number of the second kind S(n;k) occurs for two (of necessity consecutive) values of k. We prove that the number of exceptional integers less than or equal to x is O(x3=5+†), for any †> 0.
For fixed n, Stirling numbers of the second kind, S(n,r) have a single maximum.
The work of Harper and subsequent authors has shown that finite sequences (a0, …, an) arising from combinatorial problems are often such that the polynomialA(z):=∑nk=0akzkhas only real zeros. Basic examples include rows from the arrays of binomial coefficients, Stirling numbers of the first and second kinds, and Eulerian numbers. Assuming theakare nonnegative,A(1)>0 and thatA(z) is not constant, it is known thatA(z) has only real zeros iff the normalized sequence (a0/A(1), …, an/A(1)) is the probability distribution of the number of successes innindependent trials for some sequence of success probabilities. Such sequences (a0, …, an) are also known to be characterized by total positivity of the infinite matrix (ai−j) indexed by nonnegative integersiandj. This papers reviews inequalities and approximations for such sequences, calledPólya frequency sequenceswhich follow from their probabilistic representation. In combinatorial examples these inequalities yield a number of improvements of known estimates.
A technique is illustrated for finding an estimate of the Stirling numbers of the second kind, Sn, r(1 ⩽ r ⩽ n), as n → ∞, for a certain range of values of r. It is shown how the estimation can be found to any given degree of approximation. Finally, the location of the maximum of Sn, r (1 ⩽ r ⩽ n) and the value of the maximum are computed.
The distribution of the number of successes in n independent trials is "bell-shaped". The expected number of successes, say, either determines the most probable number of successes uniquely or restricts it to the pair of integers nearest to .
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