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Asymptotics of the modes of the ordered Stirling numbers
Abstract
It is known that the S(n,k) Stirling numbers as well as the ordered
Stirling numbers k!S(n,k) form log-concave sequences. Although in the first
case there are many estimations about the mode, for the ordered Stirling
numbers such estimations are not known. In this short note we study this
problem and some of its generalizations.
... Hence, OFDM-OSPM achieves the same asymptotic data rate as OFDM-SPM when K << N . On the other hand, it is known that, the value of K that maximizes K! N K satisfies K N ∼ N 2 ln 2 as N → ∞ [38]. Moreover, we have the asymptotic relationK N ! ...
... Moreover, we have the asymptotic relationK N ! N KN ∼ N !/2(ln 2) N +1 [38]. Hence, the asymptotic maximum achievable rate of the OFDM-OSPM scheme satisfies ...
In this paper, a novel modulation scheme called set partition modulation (SPM) is proposed. In this scheme, set partitioning and ordered subsets in the set partitions are used to form codewords. We define different SPM variants and depict a practical model for using SPM with orthogonal frequency division multiplexing (OFDM). For the OFDM-SPM schemes, different constellations are used to distinguish between different subsets in a set partition. To achieve good distance properties as well as better error performance for the OFDM-SPM codewords, we define a codebook selection problem and formulate such a problem as a clique problem in graph theory. In this regard, we propose a fast and efficient codebook selection algorithm. We analyze error and achievable rate performance of the proposed schemes and provide asymptotic results for the performance. It is shown that the proposed SPM variants are general schemes, which encompass multi-mode OFDM with index modulation (MM-OFDM-IM) and dual-mode OFDM with index modulation (DM-OFDM-IM) as special cases. It is also shown that OFDM-SPM schemes are capable of exhibiting better error performance and improved achievable rate than conventional OFDM, OFDM-IM, DM-OFDM-IM, and MM-OFDM-IM.
In this paper, a novel modulation scheme called set partition modulation (SPM) is proposed. In this scheme, set partitioning and ordered subsets in the set partitions are used to form codewords. We define different SPM variants and depict a practical model for using SPM with orthogonal frequency division multiplexing (OFDM). For the OFDM-SPM schemes, different constellations are used to distinguish between different subsets in a set partition. To achieve good distance properties as well as better error performance for the OFDM-SPM codewords, we define a codebook selection problem and formulate such a problem as a clique problem in graph theory. In this regard, we propose a fast and efficient codebook selection algorithm. We analyze error and achievable rate performance of the proposed schemes and provide asymptotic results for the performance. It is shown that the proposed SPM variants are general schemes, which encompass multi-mode OFDM with index modulation (MM-OFDM-IM) and dual-mode OFDM with index modulation (DM-OFDM-IM) as special cases. It is also shown that OFDM-SPM schemes are capable of exhibiting better error performance and improved achievable rate than conventional OFDM, OFDM-IM, DM-OFDM-IM, and MM-OFDM-IM.
The r-Stirling numbers of the first and second kind count restricted permutations and respectively restricted partitions, the restriction being that the first r elements must be in distinct cycles and respectively distinct subsets. The combinatorial and algebraic properties of these numbers, which in most cases generalize similar properties of the regular Stirling numbers, are explored starting from the above definition.
It is a classical result that the zeros of the Bell polynomials are real and negative. In this study we deal with the asymptotic growth of the leftmost zeros of the Bell polynomials and generalize the results for the r-Bell polynomials, too. In addition, we offer a heuristic approach for the approximation of the maximizing index of the Stirling numbers of both kind.
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.
We prove that the generating polynomial of Whitney numbers of the second kind of Dowling lattices has only real zeros.
We study some polynomials arising from Whitney numbers of the second kind of Dowling lattices. Special cases of our results include well-known identities involving Stirling numbers of the second kind. The main technique used is essentially due to Rota.
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.
We give some generating functions, recurrence relations for Whitney numbers of Dowling lattices, an explicit formula for Whitney numbers of the second kind, and other relations.
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 .
On the problem of uniqueness for the maximal Stirling number(s) of the second kind
- E R Canfield
- C Pomerance
E. R. Canfield, C. Pomerance, On the problem of uniqueness for the maximal Stirling number(s)
of the second kind, Integers 2 (2002), paper A1, 13 pp.
Kombinatorikus számokáltalánosításairól
- I Mező
I.
Mező,
Kombinatorikus
számokáltalánosításairól,
PhD
thesis
at
the
University
of
Debrecen,
2009.
Available
online
(in
Hungarian)
at
www.dea.lib.unideb.hu/dea/bitstream/handle/2437/94478/ Thesis MezoI
titkositott.pdf '