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The generalized Stirling and Bell numbers revisited
 J. Integer Seq
"... The generalized Stirling numbers Ss;h(n,k) introduced recently by the authors are shown to be a special case of the three parameter family of generalized Stirling numbers S(n,k;α,β,r) considered by Hsu and Shiue. From this relation, several properties of Ss;h(n,k) and the associated Bell numbers Bs; ..."
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The generalized Stirling numbers Ss;h(n,k) introduced recently by the authors are shown to be a special case of the three parameter family of generalized Stirling numbers S(n,k;α,β,r) considered by Hsu and Shiue. From this relation, several properties of Ss;h(n,k) and the associated Bell numbers Bs;h(n) and Bell polynomials B s;hn(x) are derived. The particular case s = 2 and h = −1 corresponding to the meromorphic Weyl algebra is treated explicitly and its connection to Bessel numbers and Bessel 1 polynomials is shown. The dual case s = −1 and h = 1 is connected to Hermite polynomials. For the general case, a close connection to the Touchard polynomials of higher order recently introduced by Dattoli et al. is established, and Touchard polynomials of negative order are introduced and studied. Finally, a qanalogue Ss;h(n,kq) is introduced and first properties are established, e.g., the recursion relation and an explicit expression. It is shown that the qdeformed numbers Ss;h(n,kq) are special cases of the typeII p,qanalogue of generalized Stirling numbers introduced by Remmel and Wachs, providing the analogue to the undeformed case (q = 1). Furthermore, several special cases are discussed explicitly, in particular, the case s = 2 and h = −1 corresponding to the qmeromorphic Weyl algebra considered by Diaz and Pariguan. 1
Topic Segmentation with a Structured Topic Model
"... We present a new hierarchical Bayesian model for unsupervised topic segmentation. This new model integrates a pointwise boundary sampling algorithm used in Bayesian segmentation into a structured topic model that can capture a simple hierarchical topic structure latent in documents. We develop an M ..."
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We present a new hierarchical Bayesian model for unsupervised topic segmentation. This new model integrates a pointwise boundary sampling algorithm used in Bayesian segmentation into a structured topic model that can capture a simple hierarchical topic structure latent in documents. We develop an MCMC inference algorithm to split/merge segment(s). Experimental results show that our model outperforms previous unsupervised segmentation methods using only lexical information on Choi’s datasets and two meeting transcripts and has performance comparable to those previous methods on two written datasets. 1
Let
, 2012
"... Using the saddle point method and multiseries { expansions, } we obtain from the generating funcn tion of the Stirling numbers of the second kind and Cauchy’s integral formula, asymptotic m results in central and noncentral regions. In the central region, we revisit the celebrated Gaussian theorem ..."
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Using the saddle point method and multiseries { expansions, } we obtain from the generating funcn tion of the Stirling numbers of the second kind and Cauchy’s integral formula, asymptotic m results in central and noncentral regions. In the central region, we revisit the celebrated Gaussian theorem with more precision. In the region m = n−n α {}, 1> α> 1/2, we analyze the dependence n
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"... In Appendix A.5 of [1] it was shown by induction that [1 − d] a−1 1 = Sd(c, t) (1) A∈Act a∈A where Sd(c, t) is a generalized Stirling number of type (−1, −d, 0) [2]. These can be computed recursively as follows: Sd(1, 1) = Sd(0, 0) = 1 (2) Sd(c, 0) = Sd(0, t) = 0 for c, t> 0 (3) Sd(c, t) = ..."
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In Appendix A.5 of [1] it was shown by induction that [1 − d] a−1 1 = Sd(c, t) (1) A∈Act a∈A where Sd(c, t) is a generalized Stirling number of type (−1, −d, 0) [2]. These can be computed recursively as follows: Sd(1, 1) = Sd(0, 0) = 1 (2) Sd(c, 0) = Sd(0, t) = 0 for c, t> 0 (3) Sd(c, t) = 0 for t> c (4) Sd(c, t) = Sd(c − 1, t − 1) + (c − 1 − dt)Sd(c − 1, t) for 0 < t ≤ c (5)
pRook Numbers and Cycle Counting in Cp ≀ Sn
"... hit numbers, wreath product. Cyclecounting rook numbers were introduced by Chung and Graham [7]. Cyclecounting qrook numbers were introduced by Ehrenborg, Haglund, and Readdy [9] and cyclecounting qhit numbers were introduced by Haglund [14]. Briggs and Remmel [4] introduced the theory of proo ..."
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hit numbers, wreath product. Cyclecounting rook numbers were introduced by Chung and Graham [7]. Cyclecounting qrook numbers were introduced by Ehrenborg, Haglund, and Readdy [9] and cyclecounting qhit numbers were introduced by Haglund [14]. Briggs and Remmel [4] introduced the theory of prook and phit numbers which is a rook theory model where the rook numbers correspond to partial permutations in Cp ≀ Sn, the wreath product of the cyclic group Cp and the symmetric group Sn, and the hit numbers correspond to signed permutations in Cp ≀ Sn. In this paper, we extend the cyclecounting qrook numbers and cyclecounting qhit numbers to the BriggsRemmel model. In such a setting, we define a multivariable version of the cyclecounting qrook numbers and cyclecounting qhit numbers where we keep track of cycles of permutations and partial permutations of Cp ≀ Sn according to the signs of the cycles. 1
Article 11.1.1 The rBell Numbers
"... The notion of rStirling numbers implies the definition of generalized Bell (or rBell) numbers. The rBell numbers have appeared in several works, but there is no systematic treatise on this topic. In this paper we fill this gap. We discuss the most important combinatorial, algebraic and analytic p ..."
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The notion of rStirling numbers implies the definition of generalized Bell (or rBell) numbers. The rBell numbers have appeared in several works, but there is no systematic treatise on this topic. In this paper we fill this gap. We discuss the most important combinatorial, algebraic and analytic properties of these numbers, which generalize similar properties of the Bell numbers. Most of these results seem to be new. It turns out that in a paper of Whitehead, these numbers appeared in a very different context. In addition, we study the socalled rBell polynomials. 1
Extensions of Spivey’s Bell number formula
"... We establish an extension of Spivey’s Bell number formula and its associated Bell polynomial extension by using HsuShiue’s generalized Stirling numbers. By means of the extension of Spivey’s Bell number formula we also extend GouldQuaintance’s new Bell number formulas. ..."
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We establish an extension of Spivey’s Bell number formula and its associated Bell polynomial extension by using HsuShiue’s generalized Stirling numbers. By means of the extension of Spivey’s Bell number formula we also extend GouldQuaintance’s new Bell number formulas.
Translated Whitney and rWhitney Numbers: A Combinatorial Approach
"... Using a combinatorial approach, we introduce the translated Whitney numbers. This seems to be more natural than to write a product of anarithmetical progression in terms of a power variable and conversely. We also extend our ideas to translated rWhitney numbers of both kinds and to translated Whitn ..."
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Using a combinatorial approach, we introduce the translated Whitney numbers. This seems to be more natural than to write a product of anarithmetical progression in terms of a power variable and conversely. We also extend our ideas to translated rWhitney numbers of both kinds and to translated WhitneyLah numbers. 1
Article Generalized qStirling Numbers and Their Interpolation Functions OPEN ACCESS
, 2013
"... axioms ..."
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