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**11 - 18**of**18**### p-Rook Numbers and Cycle Counting in Cp ≀ Sn

"... hit numbers, wreath product. Cycle-counting rook numbers were introduced by Chung and Graham [7]. Cycle-counting q-rook numbers were introduced by Ehrenborg, Haglund, and Readdy [9] and cycle-counting q-hit numbers were introduced by Haglund [14]. Briggs and Remmel [4] introduced the theory of p-roo ..."

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hit numbers, wreath product. Cycle-counting rook numbers were introduced by Chung and Graham [7]. Cycle-counting q-rook numbers were introduced by Ehrenborg, Haglund, and Readdy [9] and cycle-counting q-hit numbers were introduced by Haglund [14]. Briggs and Remmel [4] introduced the theory of p-rook and p-hit 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 cycle-counting q-rook numbers and cycle-counting q-hit numbers to the Briggs-Remmel model. In such a setting, we define a multivariable version of the cycle-counting q-rook numbers and cycle-counting q-hit numbers where we keep track of cycles of permutations and partial permutations of Cp ≀ Sn according to the signs of the cycles. 1

### Let

, 2012

"... Using the saddle point method and multiseries { expansions, } we obtain from the generating func-n tion of the Stirling numbers of the second kind and Cauchy’s integral formula, asymptotic m results in central and non-central 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 func-n tion of the Stirling numbers of the second kind and Cauchy’s integral formula, asymptotic m results in central and non-central 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

### Article 11.1.1 The r-Bell Numbers

"... The notion of r-Stirling numbers implies the definition of generalized Bell (or r-Bell) numbers. The r-Bell 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 r-Stirling numbers implies the definition of generalized Bell (or r-Bell) numbers. The r-Bell 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 so-called r-Bell 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 Hsu-Shiue’s generalized Stirling numbers. By means of the extension of Spivey’s Bell number formula we also extend Gould-Quaintance’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 Hsu-Shiue’s generalized Stirling numbers. By means of the extension of Spivey’s Bell number formula we also extend Gould-Quaintance’s new Bell number formulas.

### Article Generalized q-Stirling Numbers and Their Interpolation Functions OPEN ACCESS

, 2013

"... axioms ..."

### Topic Segmentation with a Structured Topic Model

"... We present a new hierarchical Bayesian model for unsupervised topic segmentation. This new model integrates a point-wise 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 point-wise 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

### (1 + βt) α β

"... In this section and in section 2 and 3 we will generalize the summation rule obtained by L.C. Hsu [7], involving Stirling numbers of the second kind. Given four real numbers a, b, α and β with α = 0 and β = 0, L.C. Hsu and H.Q. Yu [11], L.C. Hsu and P.J. Shiue [9] defined the symmetric Stirling-ty ..."

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In this section and in section 2 and 3 we will generalize the summation rule obtained by L.C. Hsu [7], involving Stirling numbers of the second kind. Given four real numbers a, b, α and β with α = 0 and β = 0, L.C. Hsu and H.Q. Yu [11], L.C. Hsu and P.J. Shiue [9] defined the symmetric Stirling-type pairs (〈a, b, α, β〉- pairs, for short)

### Translated Whitney and r-Whitney 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 r-Whitney 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 r-Whitney numbers of both kinds and to translated Whitney-Lah numbers. 1