Formula For Generating Function Research Articles

Combinatorial formulas for shifted dual stable Grothendieck polynomials

Abstract The K-theoretic Schur P- and Q-functions $G\hspace {-0.2mm}P_\lambda $ and $G\hspace {-0.2mm}Q_\lambda $ may be concretely defined as weight-generating functions for semistandard shifted set-valued tableaux. These symmetric functions are the shifted analogues of stable Grothendieck polynomials and were introduced by Ikeda and Naruse for applications in geometry. Nakagawa and Naruse specified families of dual K-theoretic Schur P- and Q-functions $g\hspace {-0.1mm}p_\lambda $ and $g\hspace {-0.1mm}q_\lambda $ via a Cauchy identity involving $G\hspace {-0.2mm}P_\lambda $ and $G\hspace {-0.2mm}Q_\lambda $ . They conjectured that the dual power series are weight-generating functions for certain shifted plane partitions. We prove this conjecture. We also derive a related generating function formula for the images of $g\hspace {-0.1mm}p_\lambda $ and $g\hspace {-0.1mm}q_\lambda $ under the $\omega $ involution of the ring of symmetric functions. This confirms a conjecture of Chiu and the second author. Using these results, we verify a conjecture of Ikeda and Naruse that the $G\hspace {-0.2mm}Q$ -functions are a basis for a ring.

Enumeration of non-crossing partitions according to subwords with repeated letters

In this paper, we enumerate members of the set NCn of non-crossing partitions of length n according to the number of occurrences of several infinite families of subword patterns each containing repeated letters. As a consequence of our results, we obtain explicit generating function formulas counting the members of NCn for n ? 0 according to all subword patterns of length three containing a repeated letter. In particular, we find extensions of the subword equivalences 211 ? 221, 1211 ? 1121 and 112 ? 122 over NCn.

Gaussian-bihyperbolic Numbers Containing Pell and Pell-Lucas Numbers

In this study, we define a new type of Pell and Pell-Lucas numbers which are called Gaussian-bihyperbolic Pell and Pell-Lucas numbers. We also define negaGaussian-bihyperbolic Pell and Pell-Lucas numbers. Moreover, we obtain Binet’s formulas, generating function formulas, d’Ocagne’s identities, Catalan’s identities, Cassini’s identities and some sum formulas for these new type numbers and we investigate some algebraic proporties of these. Furthermore, we give the matrix representation of Gaussian-bihyperbolic Pell and Pell-Lucas numbers.

Discrete analytic functions, structured matrices and a new family of moment problems

Using Zeilberger generating function formula for the values of a discrete analytic function in a quadrant we make connections with the theory of structured reproducing kernel spaces, structured matrices and a generalized moment problem. An important role is played by a Krein space realization result of Dijksma, Langer and de Snoo for functions analytic in a neighborhood of the origin. A key observation is that, using a simple Moebius transform, one can reduce the study of discrete analytic functions in the upper right quadrant to problems of function theory in the open unit disk. As an example, we associate to each finite positive measure on [0,2π] a discrete analytic function on the right-upper quarter plane with values on the lattice defining a positive definite function. Emphasis is put on the rational case, both when an underlying Carathéodory function is rational and when, in the positive case, the spectral function is rational. The rational case and the general case are linked via the existence of a unitary dilation, possibly in a Krein space.

Ferrers graphs, D-permutations, and surjective staircases

We introduce a new family of hyperplane arrangements inspired by the homogenized Linial arrangement (which was recently introduced by Hetyei), and show that the intersection lattices of these arrangements are isomorphic to the bond lattices of Ferrers graphs. Using recent work of Lazar and Wachs we are able to give combinatorial interpretations of the characteristic polynomials of these arrangements in terms of permutation enumeration. For certain infinite families of these hyperplane arrangements, we are able to give generating function formulas for their characteristic polynomials. To do so, we develop a generalization of Dumont’s surjective staircases, and introduce a polynomial which enumerates these generalized surjective staircases according to several statistics. We prove a recurrence for these polynomials and show that in certain special cases this recurrence can be solved explicitly to yield a generating function. We also prove refined versions of several of these results using the theory of complex hyperplane arrangements.

On avoiding 1233

In this paper, we establish a recurrence relation for finding the generating function for the number of k-ary words of length n that avoid 1233 for arbitrary k. Comparable generating function formulas may also be found counting words where a single permutation pattern of length three is avoided in addition to 1233.

Time evolution law of a two-mode squeezed light field passing through twin diffusion channels

We explore the time evolution law of a two-mode squeezed light field (pure state) passing through twin diffusion channels, and we find that the final state is a squeezed chaotic light field (mixed state) with entanglement, which shows that even though the two channels are independent of each other, since the two modes of the initial state are entangled with each other, the final state remains entangled. Nevertheless, although the squeezing (entanglement) between the two modes is weakened after the diffusion, it is not completely removed. We also highlight the law of photon number evolution. In the calculation process used in this paper, we make full use of the summation method within the ordered product of operators and the generating function formula for two-variable Hermite polynomials.

Option pricing with conditional GARCH models

This paper introduces a class of conditional GARCH models that offers significantly added flexibility to accommodate empirically relevant features of financial asset returns while admitting closed-form recursive solutions for the moment generating function, a variance dependent pricing kernel and, therefore, efficient option pricing in a realistic setting. This class of conditional GARCH models can be constructed with specifications of the GARCH dynamics and innovations, for which recursive moment generating function formulas have been derived, hence generalizing such families of models. As an example, we combine the popular Heston-Nandi model with Regime Switching to illustrate the flexibility of our methodology and demonstrate the importance in terms of option prices and Greeks of accommodating crisis periods and state dependency as well as priced variance risk.

A new decomposition of ascent sequences and Euler–Stirling statistics

As shown by Bousquet-Mélou–Claesson–Dukes–Kitaev (2010), ascent sequences can be used to encode (2+2)-free posets. It is known that ascent sequences are enumerated by the Fishburn numbers, which appear as the coefficients of the formal power series∑m=1∞∏i=1m(1−(1−t)i). In this paper, we present a novel way to recursively decompose ascent sequences, which leads to:•a calculation of the Euler–Stirling distribution on ascent sequences, including the numbers of ascents (asc), repeated entries (rep), zeros (zero) and maximal entries (max). In particular, this confirms and extends Dukes and Parviainen's conjecture on the equidistribution of zero and max.•a far-reaching generalization of the generating function formula for (asc,zero) due to Jelínek. This is accomplished via a bijective proof of the quadruple equidistribution of (asc,rep,zero,max) and (rep,asc,rmin,zero), where rmin denotes the right-to-left minima statistic of ascent sequences.•an extension of a conjecture posed by Levande, which asserts that the pair (asc,zero) on ascent sequences has the same distribution as the pair (rep,max) on (2−1)-avoiding inversion sequences. This is achieved via a decomposition of (2−1)-avoiding inversion sequences parallel to that of ascent sequences.This work is motivated by a double Eulerian equidistribution of Foata (1977) and a tempting bi-symmetry conjecture, which asserts that the quadruples (asc,rep,zero,max) and (rep,asc,max,zero) are equidistributed on ascent sequences.

Alternation acyclic tournaments

We define a tournament to be alternation acyclic if it does not contain a cycle in which descents and ascents alternate. Using a result by Athanasiadis on hyperplane arrangements, we show that these tournaments are counted by the median Genocchi numbers. By establishing a bijection with objects defined by Dumont, we show that alternation acyclic tournaments in which at least one ascent begins at each vertex, except for the largest one, are counted by the Genocchi numbers of the first kind. Unexpected consequences of our results include a pair of ordinary generating function formulas for the Genocchi numbers of both kinds and a simple model for the normalized median Genocchi numbers.

Elementary proof and application of the generating functions for generalized Hall–Littlewood functions

In this note we define a generalization of Hall–Littlewood symmetric functions using a formal group law and we give an elementary proof of the generating function formula for these generalized functions. We also present some applications of this formula.

New refined enumerations of set partitions related to sorting

ABSTRACTIn this paper, we consider a statistic defined on the sequential representation of a finite set partition which tracks the number of steps required, i.e. pushes, to put the sequence in weakly increasing order. A generating function is computed for the distribution of the statistic that records the number of pushes considered jointly with some other combinatorial parameters on the set of partitions of for each fixed n. Some further attention is paid to the distribution for the number of pushes by itself. Generating function formulas are also found for the comparable distribution on the set of non-crossing and non-nesting partitions of . As a consequence of our results, one obtains new polynomial generalizations of the Stirling, Bell and Catalan numbers.

A generalized class of restricted Stirling and Lah numbers

Abstract In this paper, we consider a polynomial generalization, denoted by u m a , b $\begin{array}{} u_m^{a,b} \end{array}$ (n, k), of the restricted Stirling numbers of the first and second kind, which reduces to these numbers when a = 1 and b = 0 or when a = 0 and b = 1, respectively. If a = b = 1, then u m a , b $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) gives the cardinality of the set of Lah distributions on n distinct objects in which no block has cardinality exceeding m with k blocks altogether. We derive several combinatorial properties satisfied by u m a , b $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) and some additional properties in the case when a = b = 1. Our results not only generalize previous formulas found for the restricted Stirling numbers of both kinds but also yield apparently new formulas for these numbers in several cases. Finally, an exponential generating function formula is derived for u m a , b $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) as well as for the associated Cauchy numbers.

Counting corners in compositions and set partitions presented as bargraphs

In this paper, we consider statistics on compositions of a fixed number and set partitions of a fixed size represented geometrically as bargraphs. By a corner of a bargraph, we mean a vertex along its polygonal boundary, where types of corners are identified by the sequence of steps directly before and after. Here, we find a generating function formula of the joint distribution for the statistics on compositions of n recording the number of corners of types uh or dh. We also find an explicit formula for the total number of corners of either type in all compositions of n, supplying both algebraic and combinatorial proofs. We then determine the joint distribution on compositions for corners of type for all i less than some fixed number as well as the comparable distributions for corners of type and on set partitions of size n having k blocks. In deriving our results, we solve various types of difference equations that are satisfied by the generating functions, using inductive arguments to determine closed form expressions.

Lecture Hall Partitions and the Affine Hyperoctahedral Group

In 1997 Bousquet-Mélou and Eriksson introduced lecture hall partitions as the inversion vectors of elements of the parabolic quotient $\widetilde{C}/C$. We provide a new view of their correspondence that allows results in one domain to be translated into the other. We determine the equivalence between combinatorial statistics in each domain and use this correspondence to translate certain generating function formulas on lecture hall partitions to new observations about $\widetilde{C}/C$.

Counting water cells in bargraphs of compositions and set partitions

In this paper, we consider statistics on compositions and set partitions represented geometrically as bargraphs. By a water cell, we mean a unit square exterior to a bargraph that lies along a horizontal line between any two squares contained within the area subtended by the bargraph. That is, if a large amount of a liquid were poured onto the bargraph from above and allowed to drain freely, then the water cells are precisely those cells where the liquid would collect. In this paper, we count both compositions and set partitions according to the number of descents and water cells in their bargraph representations and determine generating function formulas for the joint distributions on the respective structures. Comparable generating functions that count non-crossing and non-nesting partitions are also found. Finally, we determine explicit formulas for the sign balance and for the first moment of the water cell statistic on set partitions, providing both algebraic and combinatorial proofs.

A polynomial generalization of some associated sequences related to set partitions

In this paper, we consider a common polynomial generalization, denoted by $$w_m(n,k)=w_m^{a,b,c,d}(n,k)$$ , of several types of associated sequences. When $$a=0$$ and $$b=1$$ , one gets a generalized associated Lah sequence, while if $$c=0$$ , $$d=1$$ , one gets a polynomial array that enumerates a restricted class of weighted ordered partitions of size n having k blocks. The particular cases when $$a=d=0$$ and $$b=c=1$$ or when $$a=c=0$$ and $$b=d=1$$ correspond to the associated Stirling numbers of the first and second kind, respectively. We derive several combinatorial properties satisfied by $$w_m(n,k)$$ and consider further the case $$a=0$$ , $$b=1$$ . Our results not only generalize prior formulas found for the associated Stirling and Lah numbers but also yield some apparently new identities for these sequences. Finally, explicit exponential generating function formulas for $$w_m(n,k)$$ are derived in the cases when $$a=0$$ or $$b=0$$ .

Counting subwords in flattened involutions and Kummer functions

In this paper, we consider the problem of counting subword patterns in flattened involutions, which extends recent work on set partitions. We determine generating function formulas for the distribution of on the set of involutions of size n in all cases in which is a subword of length three. In the cases of 123 and 132, the exponential generating function for the distribution may be expressed in terms of the Kummer functions. In the cases of 213 and 312, we consider, instead, the ordinary generating function for the distribution and show that it satisfies a functional equation in two parameters that can be solved explicitly in the avoidance case. Recurrences are also given for the general patterns , , and , and in the first two cases, it is shown that the distribution on for the number of occurrences of the patterns in the flattened sense is P-recursive for all .

Laguerre-Polynomial-Weighted Two-Mode Squeezed State

We propose a new optical field named Laguerre-polynomial-weighted two-mode squeezed state. We find that such a state can be generated by passing the l-photon excited two-mode squeezed vacuum state Cla†lS2|00〉 through an single-mode amplitude damping channel. Physically, this paper actually is concerned what happens when both excitation and damping of photons co-exist for a two-mode squeezed state, e.g., dessipation of photon-added two-mode squeezed vacuum state. We employ the summation method within ordered product of operators and a new generating function formula about two-variable Hermite polynomials to proceed our discussion.

Evolution of l-photon excited thermo vacuum state in a single-mode damping channel

In this paper, we investigate how a kind of non-Gaussian states (l-photon excited thermo vacuum state [Formula: see text]) evolves in a single-mode damping channel. We find that it evolves into a Laguerre-polynomial-weighted real–fictitious squeezed thermo vacuum state, which exhibits strong decoherence and its original nonclassicality fades. In particular, when l = 0, in this damping process the thermo squeezing effect decreases while the fictitious-mode vacuum becomes chaotic. In overcoming the difficulty of calculation, we employ the summation method within ordered product of operators, a new generating function formula about two-variable Hermite polynomials is derived.