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# Combinatorial Interpretations of Particular Evaluations of Complete and Elementary Symmetric Functions

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## Abstract

The Jacobi-Stirling numbers and the Legendre-Stirling numbers of the first and second kind were first introduced by Everitt et al. (2002) and (2007) in the spectral theory. In this paper we note that Jacobi-Stirling numbers and Legendre-Stirling numbers are specializations of elementary and complete symmetric functions. We then study combinatorial interpretations of this specialization and obtain new combinatorial interpretations of the Jacobi-Stirling and Legendre-Stirling numbers. Keywords: Jacobi-Stirling numbers, Legendre-Stirling numbers, symmetric functions, combinatorial interpretations. 1

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... In the last years, these numbers received considerable attention especially in combinatorics and graph theory, see, for example. [1][2][3][4][5][6][7][8][9][10][11][12] CONTACT Mircea Merca mircea.merca@profinfo.edu.ro For the Jacobi-Stirling numbers of the second kind, [1] the following asymptotic statement is known: ...
... The proof follows easily taking into account that the Jacobi-Stirling numbers of the second kind are specializations of the complete homogeneous symmetric functions, [12] that is, n + k n γ = h k (2γ , 2 + 4γ , . . . , n(n − 1 + 2γ )). ...
... are well known.According toMongelli,[12] the Jacobi-Stirling numbers of the first kind are specializations of the elementary symmetric functions, that is, ...
Article
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The asymptotic behaviour of the Chebyshev–Stirling numbers of the second kind, a special case of the Jacobi–Stirling numbers, has been established in a recent paper by Gawronski, Littlejohn and Neuschel. In this paper, we provide an asymptotic formula for the Chebyshev–Stirling numbers of the first kind. New recurrence relations for the Euler–Riemann zeta function (Formula presented.) are derived in this context.
... In 2008, Andrews and Littlejohn [4] found a combinatorial interpretation of the Legendre-Stirling numbers in terms of certain generalized set partitions. In 2012, Mongelli [5] used a specialization of certain symmetric functions to develop a combinatorial interpretation of generalized Legendre-Stirling numbers that doesn't appear to directly generalize the combinatorial interpretation for Andrews and Littlejohn's Legendre-Stirling numbers. ...
... In 2012, Mongelli [5] noted that the Legendre-Stirling numbers are a specialization of certain symmetric functions and gave the following combinatorial interpretation of these numbers for any fixed polynomial with only integer roots and nonnegative coefficients, ...
... Let l be the number of blocks to the right of the block containing r m that either do not contain a labeled copy of m or contain an To form Q , we will place the 1's and the 2's in the same blocks as P . Then for 5 3 we have 1 l = , for 4 3 we have 2 l = , for 3 3 we have 3 l = , for 2 3 we have 2 l = and for 1 3 we have 2 l = . Thus our resulting Q partition that satisfies our criteria is ...
Article
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... In the last decade, the Jacobi-Stirling numbers of both kinds received considerable attention especially in combinatorics and graph theory. The application of these numbers in these areas is natural since they have many properties similar to those of the classical Stirling numbers, see, e.g., [16][17][18][19][20][21][22][23][24][25][26]. Mongelli [26] has shown that the Jacobi-Stirling numbers are specializations of the elementary and complete homogeneous symmetric functions, i.e., Due to Lema 2.1, the Jacobi-Stirling numbers of both kinds and the Bernoulli polynomials are related by ...
... The application of these numbers in these areas is natural since they have many properties similar to those of the classical Stirling numbers, see, e.g., [16][17][18][19][20][21][22][23][24][25][26]. Mongelli [26] has shown that the Jacobi-Stirling numbers are specializations of the elementary and complete homogeneous symmetric functions, i.e., Due to Lema 2.1, the Jacobi-Stirling numbers of both kinds and the Bernoulli polynomials are related by ...
Article
In this paper, we give new relationships between complete and elementary symmetric functions. These results can be used to discover and prove some identities involving r-Whitney numbers, Jacobi–Stirling numbers, Bernoulli numbers and other numbers that are specializations of complete and elementary symmetric functions.
... In fact, these numbers are a generalization of the Legendre-Stirling numbers of the second kind: it suffices to choose α = β = 0. Recently, the Jacobi-Stirling numbers and Legendre-Stirling numbers have generated a significant amount of interest from some researchers in combinatorics. It was found that the Jacobi-Stirling numbers and Legendre-Stirling numbers share many properties with the classical Stirling numbers such as similar recurrence relations, generating functions and total positivity properties, see Andrews et al. [1,2,3], Egge [12], Everitt et al. [13,14], Gelineau and Zeng [15], and Mongelli [20,21] for details. Moreover, several combinatorial interpretations of the Legendre-Stirling numbers (see Andrews and Littlejohn [3] and Egge [12] for instance) and the Jacobi-Stirling numbers (see Andrews et al. [1], Gelineau and Zeng [15], Mongelli [21] for instance) have been found. ...
... It was found that the Jacobi-Stirling numbers and Legendre-Stirling numbers share many properties with the classical Stirling numbers such as similar recurrence relations, generating functions and total positivity properties, see Andrews et al. [1,2,3], Egge [12], Everitt et al. [13,14], Gelineau and Zeng [15], and Mongelli [20,21] for details. Moreover, several combinatorial interpretations of the Legendre-Stirling numbers (see Andrews and Littlejohn [3] and Egge [12] for instance) and the Jacobi-Stirling numbers (see Andrews et al. [1], Gelineau and Zeng [15], Mongelli [21] for instance) have been found. ...
... In the last decade, the Jacobi-Stirling numbers of both kinds received considerable attention especially in combinatorics and graph theory. The application of these numbers in these areas is natural since they share many similar properties to those of the classical Stirling numbers, see, e.g., [2,3,4,7,11,12,13,19,21,22]. The second objects are the Bernoulli polynomials which are denoted in this paper by B n (x). ...
... Recently, Mongelli [22] has shown that the Jacobi-Stirling numbers are specializations of the elementary and complete homogeneous symmetric functions of the numbers 2γ, 2+4γ, . . . , n(n−1+2γ). ...
Article
Full-text available
A finite discrete convolution involving the Jacobi-Stirling numbers of both kinds is expressed in this paper in terms of the Bernoulli polynomials.
... In fact, these numbers are a generalization of the Legendre-Stirling numbers of the second kind: it suffices to choose α = β = 0. Recently, the Jacobi-Stirling numbers and Legendre-Stirling numbers have generated a significant amount of interest from some researchers in combinatorics. It was found that the Jacobi-Stirling numbers and Legendre-Stirling numbers share many properties with the classical Stirling numbers such as similar recurrence relations, generating functions and total positivity properties, see Andrews et al. [1,2,3], Egge [12], Everitt et al. [13,14], Gelineau and Zeng [15], and Mongelli [20,21] for details. Moreover, several combinatorial interpretations of the Legendre-Stirling numbers (see Andrews and Littlejohn [3] and Egge [12] for instance) and the Jacobi-Stirling numbers (see Andrews et al. [1], Gelineau and Zeng [15], Mongelli [21] for instance) have been found. ...
... It was found that the Jacobi-Stirling numbers and Legendre-Stirling numbers share many properties with the classical Stirling numbers such as similar recurrence relations, generating functions and total positivity properties, see Andrews et al. [1,2,3], Egge [12], Everitt et al. [13,14], Gelineau and Zeng [15], and Mongelli [20,21] for details. Moreover, several combinatorial interpretations of the Legendre-Stirling numbers (see Andrews and Littlejohn [3] and Egge [12] for instance) and the Jacobi-Stirling numbers (see Andrews et al. [1], Gelineau and Zeng [15], Mongelli [21] for instance) have been found. ...
Article
Full-text available
Let {Tn,k}n,k≥0 be an array of nonnegative numbers satisfying the recurrence relation Tn,k = (a1k2 + a2k + a3)Tn-1,k + (b1k2 + b2k + b3)Tn-1,k-1 with Tn,k = 0 unless 0 ≦ k ≦ n. We obtain some results for the total positivity of the matrix (Tn,k)n,k≧0, Pólya frequency properties of the row and column generating functions, and q-log-convexity of the row generating functions. This allows a unified treatment of the properties above for some triangular arrays of the second kind, including the Stirling triangle, Jacobi-Stirling triangle, Legendre-Stirling triangle, and central factorial numbers triangle.
... S(n, k; α, 0, γ)x k , n = 1, 2, 3, . . . , (5) which shows that the (horizontal) generating function, n−1 i=0 (x + γ − iα), of the n-th row sequence S(n, k; α, 0, γ) has product form. ...
... For the initial definition, elementary properties (explicit expressions, triangular recurrence relations, similarity between the Jacobi-Stirling and classical Stirling numbers, etc.) and different combinatorial interpretations of special cases of the Jacobi-Stirling numbers, see [3,4,5,6,8], respectively. ...
Article
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In this paper, we establish several properties of the unified generalized Stirling numbers of the first kind, and the Jacobi-Stirling numbers of the first kind, by means of the convolution principle of sequences. Obtained results include generalized Vandermonde convolution for the unified generalized Stirling numbers of the first kind, triangular recurrence relation for general Stirling-type numbers of the first kind, and linear recurrence formula for the Jacobi-Stirling numbers of the first kind, and so forth, thereby extending and supplementing known knowledge to the existent literature about these Stirling-type numbers.
... Since then, in analogy to the Stirling numbers, several properties of the Jacobi-Stirling numbers (or Legendre-Stirling numbers) and its companions including combinatorial interpretations have been established, [1,5,10,11,29]. ...
... Remark 4.4. By letting q → 1, the pair (Jc k n (z; q), JS k n (z; q)) 0 k n reduces to the Jacobi-Stirling numbers in [6,10,29] and the relation (4.11) reduces to [6, (4.4)]. ...
Article
We introduce, characterise and provide a combinatorial interpretation for the so-called $q$-Jacobi-Stirling numbers. This study is motivated by their key role in the (reciprocal) expansion of any power of a second order $q$-differential operator having the $q$-classical polynomials as eigenfunctions in terms of other even order operators, which we explicitly construct in this work. The results here obtained can be viewed as the $q$-version of those given by Everitt {\it et al.} and by the first author, whilst the combinatorics of this new set of numbers is a $q$-version of the Jacobi-Stirling numbers given by Gelineau and the second author.
... Recently, these numbers have attracted the attention of several authors [1][2][3][6][7][8]13,14]. In particular, a result of Egge [6,Theorem 5.1] implies that the diagonal sequences {JS(k + n, n; 1)} n≥0 and {Jc(k + n, n; 1)} n≥k are Pólya frequency sequences for any fixed k ∈ N, while Mongelli [13] studied total positivity properties of Jacobi-Stirling numbers assuming that z is a real number. ...
... As noticed by Mongelli [14], comparing with (1.1) and (1.2) one gets immediately the following identities: for n ≥ k ≥ 0, Jc(n, k; z) = e n−k (1(1 + z), 2(2 + z), . . . , (n − 1)(n − 1 + z)), (3.1) JS(n, k; z) = h n−k (1(1 + z), 2(2 + z), . . . ...
Article
Generalizing recent results of Egge and Mongelli, we show that each diagonal sequence of the Jacobi-Stirling numbers $\js(n,k;z)$ and $\JS(n,k;z)$ is a P\'olya frequency sequence if and only if $z\in [-1, 1]$ and study the $z$-total positivity properties of these numbers. Moreover, the polynomial sequences $$\biggl\{\sum_{k=0}^n\JS(n,k;z)y^k\biggr\}_{n\geq 0}\quad \text{and} \quad \biggl\{\sum_{k=0}^n\js(n,k;z)y^k\biggr\}_{n\geq 0}$$ are proved to be strongly $\{z,y\}$-log-convex. In the same vein, we extend a recent result of Chen et al. about the Ramanujan polynomials to Chapoton's generalized Ramanujan polynomials. Finally, bridging the Ramanujan polynomials and a sequence arising from the Lambert $W$ function, we obtain a neat proof of the unimodality of the latter sequence, which was proved previously by Kalugin and Jeffrey.
... [3], [17]. During the past decade they received considerable attention resulting in a series of papers on differential equations, combinatorics, and graph theory [1], [2], [3], [7], [8], [10], [12], [13], [17], [18], [24], [26], [27]. A brief account of the background is given in section 2 below. ...
... Starting with the Legendre-Stirling case γ = 1 in [1], which corresponds to the parameters α = β = 0, in the sequel a series of authors developed various combinatorial models, thereby illustrating the significance of the Jacobi-Stirling numbers and related quantities. We refer to the articles [2], [3], [7], [8], [10], [17], [18], [26], [27]. However none of these papers contains information on asymptotics. ...
Article
Full-text available
For the classical Stirling numbers of the second kind, asymptotic formulae are derived in terms of a local central limit theorem. The underlying probabilistic approach also applies to the Chebyshev-Stirling numbers, a special case of the Jacobi-Stirling numbers. Essential features are uniformity properties and the fact that the leading terms of the asymptotics are given explicitly and they contain elementary expressions only. Thereby supplements of the asymptotic analysis of these numbers are established.
... 1.4], r -Jacobi-Stirling [5,Sect. 1.4], two classes of p-Stirling numbers of the second kind [11] and other numbers studied in [9]. More generally, let n, s 1 , ..., s n be positive integers and (exp v n (t)) k exp u n (γ t) , n ≥ 1, k ≥ 0, are three sequences of non-negative integers with s n ≥ 1 for all n and exp n (t) := 1 + t + · · · + t n n! . ...
Article
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By counting on the multiset partitions, we give in this paper a combinatorial interpretation of a class of the generalized Stirling numbers generalizing the Stirling, r-Stirling, Jacobi–Stirling, r-Jacobi–Stirling and two classes of p-Stirling numbers of the second kind.
... Mongelli [23] has shown that the Jacobi-Stirling numbers of the second kind are specializations of the complete symmetric functions, that is JS (k) n (z) = h n−k (0, 1 + z, . . . , k(k + z)). ...
Article
Full-text available
By the LU factorization, we evaluate a determinant involving the complete symmetric functions. From a viewpoint of symmetric functions, some results for the evaluations of the determinants of the matrices consisting of the numbers of the Stirling-type are given.
... We let n k γ denote a Jacobi-Stirling number of the first kind, and n k γ a Jacobi-Stirling number of the second kind. The article [11] proves the equalities (4.5) e k (2γ, 2 + 4γ, . . . , n(n − 1 + 2γ)) = n k γ and (4.6) h k (2γ, 2 + 4γ, . . . ...
Preprint
We present a generalization of the Newton-Girard identities, along with some applications. As an addendum, we collect many evaluations of symmetric polynomials to which these identities apply.
... In particular, Jc k n (1) = Lc (n, k). Properties and combinatorial interpretations of the Jacobi-Stirling numbers of both kinds were extensively studied in [1,10,11,12,15,16,17]. The Jacobi-Stirling numbers share many similar properties to those of the Stirling numbers. ...
Preprint
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The Legendre-Stirling numbers of the second kind were introduced by Everitt et al. in the spectral theory of powers of the Legendre differential expressions. In this paper, we provide a combinatorial code for Legendre-Stirling set partitions. As an application, we obtain combinatorial expansions of the Legendre-Stirling numbers of both kinds. Moreover, we present grammatical descriptions of the Jacobi-Stirling numbers of both kinds.
... Recall that the Jacobi-Stirling numbers were discovered in 2007 as a result of a problem involving the spectral theory of powers of the classical second-order Jacobi differential expression. In the last years, these numbers received considerable attention especially in combinatorics and graph theory, see, e.g., [2][3][4][11][12][13][14][15]18,19,22,27,28]. Throughout the article, the symbol ∼ means asymptotic equivalence, i.e., we write a n ∼ b n when lim n→∞ a n b n = 1 (we also use ∼ for the asymptotic equivalence of functions). ...
Article
Full-text available
In this paper, we give asymptotic formulas that combine the Euler-Riemann zeta function and the Chebyshev-Stirling numbers of the first kind. These results allow us to prove an asymptotic formula related to the $n$th complete homogeneous symmetric function, which was recently conjectured by the second author: $$h_{n}\left(1,\left( \frac{k}{k+1}\right)^2 ,\left( \frac{k}{k+2} \right)^2 ,\ldots \right) \sim \binom{2k}{k}\quad\text{as}\quad n\to\infty.$$A direct proof of this asymptotic formula, due to Gerg\H{o} Nemes, is provided in the appendix.
... The Jacobi-Stirling numbers were discovered in 2007 as a result of a problem involving the spectral theory of powers of the classical second-order Jacobi differential expression. In the last decade, these numbers received considerable attention especially in combinatorics and graph theory, see, e.g., [2,3,4,8,9,10,11,12,15,16,17,19,20,21,22]. In this paper, we shall prove the following inequalities. ...
Article
In this paper, we give an infinite sequence of inequalities involving the Riemann zeta function with even arguments ? (2n) and the Chebyshev-Stirling numbers of the first kind. This result is based on a recent connection between the Riemann zeta function and the complete homogeneous symmetric functions [18]. An interesting asymptotic formula related to the nth complete homogeneous symmetric function is conjectured in this context. hn (1, ( k k+1 )2 , ( k k+2 )2 , . . . ) - ( 2k k ) , n→8.
... Since the generalized Stirling numbers do not satisfy similar two term recurrences, they are not evaluations of elementary symmetric polynomials. Other generalizations of the Stirling number are, however, symmetric polynomial evaluations - [14] uses this technique of matching recurrence relations to establish that the Jacobi-Stirling numbers are in fact symmetric polynomial evaluations. This is important because it means that there is no simple product factorization of the generating function for the Stirling numbers. ...
Article
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We introduce new refinements of the Bell, factorial, and unsigned Stirling numbers of the first and second kind that unite the derangement, involution, associated factorial, associated Bell, incomplete Stirling, restricted factorial, restricted Bell, and $r$-derangement numbers (and probably more!). By combining methods from analytic combinatorics, umbral calculus, and probability theory, we derive several recurrence relations and closed form expressions for these numbers. By specializing our results to the classical case, we recover explicit formulae for the Bell and Stirling numbers as sums over compositions.
... The Jacobi-Stirling numbers were discovered in 2007 as a result of a problem involving the spectral theory of powers of the classical second-order Jacobi differential expression. In the last decade, these numbers received considerable attention especially in combinatorics and graph theory, see, e.g., [2,3,4,8,9,10,11,12,15,16,17,19,20,21,22]. In this paper, we shall prove the following inequalities. ...
Article
In this paper, we give an infinite sequence of inequalities involving the Riemann zeta function with even arguments and the Chebyshev-Stirling numbers of the first kind. This result is based on a recent connection between the Riemann zeta function and the complete homogeneous symmetric functions. An interesting asymptotic formula related to the n-th complete homogeneous symmetric function is conjectured in this context.
... These numbers received considerable attention especially in combinatorics and graph theory. The application of the Jacobi-Stirling numbers in these areas is natural since they have many properties similar to those of the classical Stirling numbers, see, e.g., [1][2][3]6,7,9,10,[12][13][14][15]. It is well know that the unnormalized sinc function can be defined analytically by the infinite sum [8], ...
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Accurate approximations in terms of the Chebyshev–Stirling numbers of the first kind are established in the paper for the cardinal sine function. Similar results are presented for the hyperbolic cardinal sine function.
... They count the number of ways to partition a set of n objects into k non-empty subsets. [2], the Stirling numbers are specializations of elementary and complete symmetric functions. This allows us to consider the following two identities of formal power series in t for the Stirling num- bers ...
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In this note, the author proves that sums of powers of the first n positive integers can be expressed as finite discrete convolutions.
... In this paper, we show that these experimental results are very special cases of more general identities. In fact, it is well-known [4,5] that the unsigned Stirling numbers of the first kind are the elementary symmetric functions of the numbers 1, 2, . . . , n, i.e., ...
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The complete and elementary symmetric functions are special cases of Schur functions. It is well-known that the Schur functions can be expressed in terms of complete or elementary symmetric functions using two determinant formulas: Jacobi–Trudi identity and Nägelsbach–Kostka identity. In this paper, we study new connections between complete and elementary symmetric functions.
... The Legendre differential operator corresponds to the case α = β = 0, i.e., γ = 1, and hence the numbers n j 1 are called Legendre-Stirling numbers of the second kind. Besides the already mentioned original field of differential equations during the past decade the Jacobi-Stirling numbers received considerable attention especially in combinatorics and graph theory, see, e.g., [1], [2], [3], [6], [7], [9], [15], [16], [17], [21], [22], [23]. Among the n j γ 's the Legendre-Stirling numbers n j 1 ...
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For the Legendre-Stirling numbers of the second kind asymptotic formulae are derived in terms of a local central limit theorem. Thereby, supplements of the recently published asymptotic analysis of the Chebyshev-Stirling numbers are established. Moreover, we provide results on the asymptotic normality and unimodality for modified Legendre-Stirling numbers.
... The identities presented in these corollaries are just some of the consequences of Theorem 1. Connections between r-Stirling numbers and Whitney numbers, Whitney numbers and Stirling numbers, or r-Stirling numbers and Stirling numbers can be immediately derived from our theorem. According to [2,3,12,13,19], the similar identities involving Legendre-Stirling numbers, Jacobi-Stirling numbers and central factorial numbers with odd indices can be obtained as well. ...
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The complete and elementary symmetric functions are specializations of Schur functions. In this paper, we use this fact to give two identities for the complete and elementary symmetric functions. This result can be used to proving and discovering some algebraic identities involving r-Whitney ando ther special numbers.
... Furthermore, the Stirling numbers, restricted Stirling numbers, Legendre-Stirling numbers, Jacobi- Stirling numbers and central factorial numbers of both kinds are specializations of elementary and complete symmetric functions. More details can be found in [4, 13]. Therefore, some combinatorial identities and its -analogs can be easily derived. ...
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The nth-order determinant of a Toeplitz-Hessenberg matrix is expressed as a sum over the integer partitions of n. Many combinatorial identities involving integer partitions and multinomial coefficients can be generated using this formula.
... Recently, Mongelli [10] has shown that the Jacobi-Stirling numbers are specializations of the elementary and complete homogeneous symmetric functions, i.e., n n − k γ = e k (2γ, 2 + 4γ, . . . , (n − 1)(n − 2 + 2γ)) and n + k n γ = h k (2γ, 2 + 4γ, . . . ...
Article
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The Jacobi–Stirling numbers of both kinds are specializations of elementary and complete homogeneous symmetric functions. We use this fact to discover and prove some algebraic identities involving Jacobi–Stirling numbers. The Legendre–Stirling numbers are very special cases of the Jacobi–Stirling numbers. New connections between the Legendre–Stirling numbers and the central factorial numbers of odd indices are presented.
... We show below, in Theorem 3.1(iii), that the Jacobi-Stirling numbers satisfy a certain triangular recurrence relation that allows for a fast computation of these numbers. As with the classical Stirling numbers there occur, in a natural way, corresponding (signless) Jacobi-Stirling numbers n j γ of the first kind (see Section 5 below; see also [7], [11], and [16]). In view of this, we occasionally will call n j γ Jacobi-Stirling numbers of the second kind. ...
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The Jacobi-Stirling numbers were discovered as a result of a problem involving the spectral theory of powers of the classical second-order Jacobi differential expression. Specifically, these numbers are the coefficients of integral composite powers of the Jacobi expression in Lagrangian symmetric form. Quite remarkably, they share many properties with the classical Stirling numbers of the second kind which, as shown in LW, are the coefficients of integral powers of the Laguerre differential expression. In this paper, we establish several properties of the Jacobi-Stirling numbers and its companions including combinatorial interpretations thereby extending and supplementing known contributions to the literature of Andrews-Littlejohn, Andrews-Gawronski-Littlejohn, Egge, Gelineau-Zeng, and Mongelli.
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The Legendre-Stirling numbers were discovered in 2002 as a result of a problem in- volving the spectral theory of powers of the classical second-order Legendre dierential expression. Speci…cally, these numbers are the coe¢ cients of integral composite powers of the Legendre expres- sion in Lagrangian symmetric form. Quite remarkably, they share many similar properties with the classical Stirling numbers of the second kind which, as shown in (9), are the coe¢ cients of integral powers of the Laguerre dierential expression. An open question, regarding the Legendre-Stirling numbers, has been to obtain a combinatorial interpretation of these numbers. In this paper, we provide such an interpretation.
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The Legendre–Stirling numbers are the coefficients in the integral Lagrangian symmetric powers of the classical Legendre second-order differential expression. In many ways, these numbers mimic the classical Stirling numbers of the second kind which play a similar role in the integral powers of the classical second-order Laguerre differential expression. In a recent paper, Andrews and Littlejohn gave a combinatorial interpretation of the Legendre–Stirling numbers. In this paper, we establish several properties of the Legendre–Stirling numbers; as with the Stirling numbers of the second kind, they have interesting generating functions and recurrence relations. Moreover, there are some surprising and intriguing results relating these numbers to some classical results in algebraic number theory.
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The Jacobi-Stirling numbers of the first and second kinds were introduced in 2006 in the spectral theory and are polynomial refinements of the Legendre-Stirling numbers. Andrews and Littlejohn have recently given a combinatorial interpretation for the second kind of the latter numbers. Noticing that these numbers are very similar to the classical central factorial numbers, we give combinatorial interpretations for the Jacobi-Stirling numbers of both kinds, which provide a unified treatment of the combinatorial theories for the two previous sequences and also for the Stirling numbers of both kinds. Comment: 15 pages
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Introduction In studying representability of matroids, Lindstrom [42] gave a combinatorial interpretation to certain determinants in terms of disjoint paths in digraphs. In a previous paper [25], the authors applied this theorem to determinants of binomial coe#cients. Here we develop further applications. As in [25], the paths under consideration are lattice paths in the plane. Our applications may be divided into two classes: first are those in which a determinant is shown to count some objects of combinatorial interest, and second are those which give a combinatorial interpretation to some numbers which are of independent interest. In the first class are formulas for various types of plane partitions, and in the second class are combinatorial interpretations for Fibonomial coe#cients, Bernoulli numbers, and the less-known Salie and Faulhaber numbers (which arise in formulas for sums of powers, and are closely related to Genocchi and Bernoulli numbers). Other enumerative appl
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We develop the left-definite analysis associated with the self-adjoint Jacobi operator , generated from the classical second-order Jacobi differential expressionin the Hilbert space , where w[alpha],[beta](t)=(1-t)[alpha](1+t)[beta], that has the Jacobi polynomials as eigenfunctions; here, [alpha],[beta]>-1 and k is a fixed, non-negative constant. More specifically, for each , we explicitly determine the unique left-definite Hilbert-Sobolev space and the corresponding unique left-definite self-adjoint operator in associated with the pair . The Jacobi polynomials form a complete orthogonal set in each left-definite space and are the eigenfunctions of each . Moreover, in this paper, we explicitly determine the domain of each as well as each integral power of . The key to determining these spaces and operators is in finding the explicit Lagrangian symmetric form of the integral composite powers of l[alpha],[beta],k[[middle dot]]. In turn, the key to determining these powers is a double sequence of numbers which we introduce in this paper as the Jacobi-Stirling numbers. Some properties of these numbers, which in some ways behave like the classical Stirling numbers of the second kind, are established including a remarkable, and yet somewhat mysterious, identity involving these numbers and the eigenvalues of .
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The Jacobi–Stirling numbers of the first and second kinds were first introduced in Everitt et al. (2007) [8] and they are a generalization of the Legendre–Stirling numbers. Quite remarkably, they share many similar properties with the classical Stirling numbers. In this paper we study total positivity properties of these numbers. In particular, we prove that the matrix whose entries are the Jacobi–Stirling numbers is totally positive and that each row and each column is a Pólya frequency sequence, except for the columns with (unsigned) numbers of the first kind.
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We propose a weighting of set partitions which is analogous to the major index for permutations. The corresponding weight generating function yields the q-Stirling numbers of the second kind of Carlitz and Gould. Other interpretations of maj are given in terms of restricted growth functions, rook placements and reduced matrices. The Foata bijection interchanging inv and maj for permutations also has a version for partitions. Finally, we generalize these constructions to an analog of Rawling's rmaj and to two new kinds of p, q-Stirling numbers. 1. THE MAJOR INDEX OF A p ARTITION Versions of the q-Stirling numbers of the second kind were first introduced by Carlitz [1, 2] and Gould [7]. Later, Milne [10] showed that those of the second kind could be viewed combinatorially as generating functions for an inversion statistic on partitions. It is well known that the q-binomial coefficients describe the distribution of two statistics on permutations: the inversion number (inv) and the major index (maj). Thus it is natural to hope for an analog of maj for partitions. The purpose of this paper is to describe such an analog and some of its properties. The rest of this section is devoted to basic definitions. In Sections 2 and 3 we will discuss other interpretations of the major index in terms of restricted growth functions, rook placements and reduced matrices, as has been done by Milne [10], Garsia and Remmel [5], Wachs and White [13] and Leroux [8] for various versions of the inversion number. Since both inv and maj have the same distribution, there should be a bijection interchanging the two. Foata [3] gave such a map for permutations and we present the partition analog in Section 4. Next, we generalize both statistics using Rawlings' rmaj [11]. Section 6 considers joint distributions, yielding two new kinds of p, q-Stirling numbers. Finally, we end with some comments. Let !J = {l, 2, ... , n }. The set of all partitions of !J into k disjoint subsets or blocks will be denoted S(!J, k). Thus the ordinary Stirling numbers of the second kind are S(n, k) = IS(!J, k)I, where I· I denotes cardinality. The blocks of ;r E S(!J, k) will be written as capital letters separated by slashes, while elements of the blocks will be set in lower case. Furthermore, we will always put ;r = B ii B2 / • • • I Bk in standard form with min B 1 <min B2 <···<min Bk.
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We develop the left-definite analysis associated with the self-adjoint Jacobi operator A (,) k , generated from the classical second- order Jacobi differential expression
Book
A classical monograph on symmetric functions.
Article
It was first observed in (F. Brenti, Mem. Amer. Math. Soc.413, 1989) that Pólya frequency sequences arise often in combinatorics. In this work we point out that the same is true, more generally, for totally positive matrices and that there is an intimate connection between them and some generalizations of the classical symmetric functions.
Article
In this paper, we develop the left-definite spectral theory associated with the self-adjoint operator A(k) in L2(−1,1), generated from the classical second-order Legendre differential equationthat has the Legendre polynomials {Pm(t)}m=0∞ as eigenfunctions; here, k is a fixed, nonnegative constant. More specifically, for k>0, we explicitly determine the unique left-definite Hilbert–Sobolev space Wn(k) and its associated inner product (·,·)n,k for each Moreover, for each we determine the corresponding unique left-definite self-adjoint operator An(k) in Wn(k) and characterize its domain in terms of another left-definite space. The key to determining these spaces and inner products is in finding the explicit Lagrangian symmetric form of the integral composite powers of ℓL,k[·]. In turn, the key to determining these powers is a remarkable new identity involving a double sequence of numbers which we call Legendre–Stirling numbers.
Article
We prove the q-log-concavity of the q-Stirling numbers of the second kind, which was recently conjectured by Lynne Butler, by suitably extending her injective proof of the analogous property of the q-binomial coefficients. For this we introduce new combinatorial interpretations of Stirling numbers of both kinds in terms of “0–1 tableaux” inspired from a row-reduced echelon matrix representation of restricted growth functions. Other related results, methods, counterexamples, and conjectures are discussed.
Article
We first give a combinatorial interpretation of Everitt, Littlejohn, and Wellman’s Legendre–Stirling numbers of the first kind. We then give a combinatorial interpretation of the coefficients of the polynomial (1−x)3k+1∑n=0∞{{n+kn}}xn analogous to that of the Eulerian numbers, where {{nk}}are Everitt, Littlejohn, and Wellman’s Legendre–Stirling numbers of the second kind. Finally we use a result of Bender to show that the limiting distribution of these coefficients as n approaches infinity is the normal distribution.
• L Carlitz
Carlitz L. On abelian fields Trans. Amer. Math. Soc., 35 (1933), pp. 122-136.