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Stirling numbers and Bessel numbers have a long history, and both have been generalized in a variety of directions. Here, we present a second level generalization that has both as special cases. This generalization often preserves the inverse relation between the first and second kind, and has simple combinatorial interpretations. We also frame the discussion in terms of the exponential Riordan group. Then the inverse relation is just the group inverse, and factoring inside the group leads to many results connecting the various Stirling and Bessel numbers.

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... (8) The r-Bell numbers B n,r count the total number of r-partitions of [n + r]. (9) The r-Fubini numbers F n,r count the total number of ordered r-partitions of [n + r]. (10) The restricted Stirling numbers n k ≤m count the number of partitions of [n] into k subsets with blocks of size at most m. ...

... The valuation of B n are discussed in Amdeberhan et al. [1]. For the prime p = 2, it is shown that In the case p = 3, experimental data shows that ν 3 (B n ) = 0 unless n ≡ {4, 8,9,11,17,21,22, 24} mod 26. Also ν 3 (B n ) = 1 if n ≡ 9, 11, 22, 24 mod 26 and that ν 3 (B n ) has a valuation tree structure for n ≡ 4, 8, 17, 21 mod 26. ...

... These sequences are discussed in [40]. Further information about these numbers may be found in [6,9,26]. ...

Extensions of a set partition obtained by imposing bounds on the size of the parts is examined. Arithmetical and combinatorial properties of a variety of such counting sequences appearing by this.

... Recently, Cheon et al. [12] introduced a generalization of this sequence called r-Bessel numbers B r (n, k). This new sequence counts the number of set partitions of [n + r] := {1, 2, . . . ...

... There is a combinatorial formula to evaluate the r-Bessel numbers (cf. [12]): ...

... The r-Bessel numbers can be defined by means of Riordan arrays [12] as follows [B r (n, k)] = (1 + z) r , z + z 2 2 , where (, ) denoted an element of an exponential Riordan group. For example, if r = 2 we have the following array: ...

In the present article we establish some combinatorial properties involving r-Bessel numbers of the second kind. These identities are deduced from the combinatorial interpretation by using restricted set partitions. Additionally, we introduce the restricted r-Bell numbers in analogy to the well-known Bell numbers. We derive several recurrence relations, combinatorial sums, arithmetical properties (2-adic valuation) and asymptotic results for these sequences.

... Recently, Cheon et al. [13] The special elements are highlighted. The exponential generating function of restricted r-Stirling numbers n k ≤m,r is given by ...

... This new sequence was studied by Cheon et al. [13] using Riordan arrays. In particular, if r = 0 we obtain the number of involutions of the n elements, denoted in [2] by Inv 1 (n). ...

... In [13] the authors also introduced r-Bessel numbers of the first kind, V r (n, k), by the following generating function: ...

In this paper we introduce restricted r-Stirling numbers of the first kind. Together with restricted r-Stirling numbers of the second kind and the associated r-Stirling numbers of both kinds, by giving more arithmetical and combinatorial properties, we introduce a new generalization of incomplete poly-Cauchy numbers of both kinds and incomplete poly-Bernoulli numbers.

... the (unsigned) first-kind ones. For an (n + r)-set, B (r) n,k counts partitions with each block having size 1 or 2, subject to the restriction that r distinguished elements must be placed in distinct blocks [52]. This interpretation provides a proof that S n,k (1, 2; r) (when r ∈ N) is nonzero if and only if 0 n − k n+r 2 . ...

Many combinatorial and other number triangles are solutions of recurrences of the Graham-Knuth-Patashnik (GKP) type. Such triangles and their defining recurrences are investigated analytically. They are acted on by a transformation group generated by two involutions: a left-right reflection and an upper binomial transformation, acting row-wise. The group also acts on the bivariate exponential generating function (EGF) of the triangle. By the method of characteristics, the EGF of any GKP triangle has an implicit representation in terms of the Gauss hypergeometric function. There are several parametric cases when this EGF can be obtained in closed form. One is when the triangle elements are the generalized Stirling numbers of Hsu and Shiue. Another is when they are generalized Eulerian numbers of a newly defined kind. These numbers are related to the Hsu-Shiue ones by an upper binomial transformation, and can be viewed as coefficients of connection between polynomial bases, in a manner that generalizes the classical Worpitzky identity. Many identities involving these generalized Eulerian numbers and related generalized Narayana numbers are derived, including closed-form evaluations in combinatorially significant cases.

The previous chapters have shown that the theory of Riordan arrays is a powerful tool for studying combinatorial sums and special polynomial and number sequences. One of the well-known classes of polynomial sequences is the class of Sheffer sequences, including many important sequences such as Bernoulli polynomials, Euler polynomials, Abel polynomials, Hermite polynomials, Laguerre polynomials, etc. This class contains the subclasses of associated sequences and Appell sequences. In [57–59], Rota, Roman, et al. established a solid background for Sheffer sequences by using the theory of modern umbral calculus and finite operator calculus. In [56], Roman further developed the theory of umbral calculus and generalized the concept of Sheffer sequences so that more special polynomial sequences are included such as the sequences of Gegenbauer polynomials, Chebyshev polynomials, and Jacobi polynomials. Using Roman’s notations, a generalized Sheffer sequence (sn(x))n∈N is defined by a generating function of the form 6.0.1A(t)εx(B(t))=∑k=0∞sk(x)cktk,where εx(t)=∑k=0∞xktk/ck is a generalization of the exponential series, and (ck)k≥0 is a non-zero sequence with c0=1. When ck=k!, (6.0.1) defines the (exponential) Sheffer sequence and its generating function turns into A(t)exB(t). Note that there are several similar names presented in the literature, such as sequences of Sheffer A-type zero [63, 64] and generalized Appell sequences [9–12]. The connection between Riordan arrays and Sheffer sequences has already been pointed out by Shapiro et al. [62] and Sprugnoli [20, 65, 66]. In fact, the classical Riordan arrays are related to the 1-umbral calculus, and thus related to Sheffer sequences defined by (6.0.1) with ck=1, and the exponential Riordan arrays shown in Sect. 6.1 are related to (exponential) Sheffer sequences with ck=k!. In other words, the exponential Riordan arrays are related to the k!-umbral calculus. In this chapter, we introduce the theory of exponential Riordan arrays and their production matrices in Sect. 6.1. Part of this section comes from Deutsch and Shapiro [26] and Barry [7, 8]. In Sects. 6.2 and 6.3, we introduce the concept of generalized Riordan arrays, and give explicitly the relationships between the generalized Riordan arrays and generalized Sheffer sequences. Then, we present the important properties and applications of the generalized Riordan arrays. Furthermore, the determinantal definition for Sheffer sequences using the relations between Riordan arrays and Sheffer sequences is also given. In Sect. 6.4, we introduce some important special Riordan arrays and Sheffer sequences. We will also give the basic properties of these arrays and polynomial sequences by using the results obtained in the previous two sections. Finally, in Sect. 6.5, we present the double Riordan arrays and their related Sheffer polynomial sequence pairs as well as their applications in combinatorics and series summations. Readers can also refer to the further discussion on the (exponential) Sheffer sequence and the classical Riordan array in He et al. [40] and the discussion on the generalized Sheffer sequence and the generalized Riordan array in Gould et al. [29], He [31–33], Wang [71], and Wang et al. [73].

We discuss the use of the replica ansatz in computing free energies in random matrix theory and confirm a conjectured condition on analytic continuation in the replica index at large-N.

We discuss the use of the replica ansatz in computing free energies in random matrix theory, and confirm a conjectured condition on analytic continuation in the replica index at large-N.

In this paper, we show that an infinite lower triangular matrix A=[aij]i,j∈N0 is an exponential Riordan matrix A=E(g,f) given by ∑i≥jaijzⁱ/i!=gfj/j! if and only if there exist both a horizontal pair {hn;h˜n}n≥0 and a vertical pair {vn;v˜n}n≥0 of sequences that represent all the elements in the matrix. As a consequence, we obtain that if the horizontal and vertical pairs of an exponential Riordan matrix are identical then the matrix is an involution. In addition, this concept can be applied to obtain the determinants of the production matrix and some conditions for the d-orthogonality of the Sheffer polynomial sequences.

Recently, the authors developed a q-analogue for Riordan matrices by means of Eulerian generating functions of the form where is the q-factorial. We apply this concept to give q-analogues of some familiar objects from the set partitions with double restrictions on blocks, namely the -Bessel numbers of both types. By setting and letting , these numbers may be reduced to the q-Stirling numbers of both kinds. Several algebraic formulas for the q-analogues are also derived using combinatorial methods together with the concept of q-Riordan matrices. In particular, q-analogues of the classical Bessel numbers of both kinds and their combinatorial interpretations are obtained.

4] P. Barry. A Catalan transform and related transformations on integer sequences. J. of Integer Sequences, 8 (2005) 05.4.5. [5] P. Barry. On a family of generalized Pascal triangles defined by exponential Riordan arrays. J. of Integer Sequences, 10 (2007) 07.3.5. [6] P. Barry. Some observations on the Lah and Laguerre transforms of integer sequences. J. of Integer Sequences, 10 (2007) 07.4.6. [7] P. Barry. A note on Krawtchouk polynomials and Riordan arrays. J. of Integer Sequences, 11 (2008) 08.2.2. [8] P. Barry, and P. Fitzpatrick. On a one-parameter family of Riordan arrays and the weight distribution of MDS codes.

We show that the generating function of the symmetric group with respect to five particular statistics gives rise to an exponential Riordan array, whose inverse is the coefficient array of the associated orthogonal polynomials. This also provides us with an LDU factorization of the Hankel matrix of the associated moments.

The so-called multi-restricted numbers generalize and extend the role of Stirling numbers and Dessel numbers in various problems of combinatorial enumeration. Multi-restricted numbers of the second kind count set partitions with a given number of parts, none of whose cardinalities may exceed a fixed threshold or "restriction". The numbers are shown to satisfy a three-term recurrence relation. Both analytic and combinatorial proofs for this relation are presented. Multi-restricted numbers of both the first and second kinds provide connections between the orbit decompositions of subsets of powers of a finite group permutation representation, in which the number of occurrences of elements is restricted. An exponential generating function for the number of orbits on such restricted powers is given in terms of powers of partial sums of the exponential function.

In this paper, using exponential Riordan arrays, we investigate the Bessel numbers and Bessel matrices. By exploring links between the Bessel matrices, the Stirling matrices and the degenerate Stirling matrices, we show that the Bessel numbers are special case of the degenerate Stirling numbers, and derive explicit formulas for the Bessel numbers in terms of the Stirling numbers and binomial coefficients.

Historically, there exist two versions of the Riordan array concept. The older one (better known as recursive matrix) consists of bi-infinite matrices (dn,k)n,k∈Z(dn,k)n,k∈Z (k>nk>n implies dn,k=0dn,k=0), deals with formal Laurent series and has been mainly used to study algebraic properties of such matrices. The more recent version consists of infinite, lower triangular arrays (dn,k)n,k∈N(dn,k)n,k∈N (k>nk>n implies dn,k=0dn,k=0), deals with formal power series and has been used to study combinatorial problems. Here we show that every Riordan array induces two characteristic combinatorial sums in three parameters n,k,m∈Zn,k,m∈Z. These parameters can be specialized and generate an indefinite number of other combinatorial identities which are valid in the bi-infinite realm of recursive matrices.

A General telephone exchange with N subscribers is indicated schematically in Fig. 1. The basic function of an exchange is that of setting up a connection between any pair of subscribers. In operation the exchange must “remember,” in some form, which subscribers are connected together until the corresponding calls are completed. This requires a certain amount of internal memory, depending on the number of subscribers, the maximum calling rate, etc. A number of relations will be derived based on these considerations which give the minimum possible number of relays, crossbar switches or other elements necessary to perform this memory function. Comparison of any proposed design with the minimum requirements obtained from the relations gives a measure of the efficiency in memory utilization of the design.

The concept of a Riordan array is used in a constructive way to find the generating function of many combinatorial sums. The generating function can then help us to obtain the closed form of the sum or its asymptotic value. Some examples of sums involving binomial coefficients and Stirling numbers are examined, together with an application of Riordan arrays to some walk problems.

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.

An investigation is made of the polynomials fk(n) = S(n + k, n) and gk(n) = (−1)ks(n, n − k), where S and s denote the Stirling numbers of the second and first kind, respectively. The main result gives a combinatorial interpretation of the coefficients of the polynomial (1 − x)2k+1Σn=0∞fk(n)xn analogous to the well-known combinatorial interpretation of the Eulerian numbers in terms of descents of permutations.

Multirestricted Stirling numbers of the second kind count the number of partitions of a given set into a given number of parts, each part being restricted to at most a fixed number of elements. Multirestricted numbers of the first kind are then defined as elements of the matrix inverse to the matrix of corresponding multirestricted numbers of the second kind. The anomalous sign behavior of these latter numbers makes them impervious to combinatorial analysis. In answer to a conjecture that has remained open for several years, we derive a reciprocity law for multirestricted Stirling numbers using algebraic techniques based on polynomial recursions. As corollaries, we obtain new recurrence relations for multirestricted numbers, and a new algebraic derivation of the reciprocity law for Stirling numbers.

A Walk through Combinatorics

- M Bóna

M. Bóna, A Walk through Combinatorics, second ed., World Scientific, Singapore, 2006.