Combinatorial Interpretation Research Articles

A q-analogue of generalized translated Whitney numbers: Cigler's approach

Using a certain combinatorial interpretation in terms of set partition, a \(q\)-analogue of generalized translated Whitney numbers of the second kind is defined in this paper. Some properties such as the recurrence relation, explicit formula, and certain symmetric formula are obtained. Moreover, a \(q\)-analogue of generalized translated Whitney numbers of the first kind is introduced to obtain a \(q\)-analogue of the orthogonality and inverse relations of the two kinds of generalized translated Whitney numbers.

On the Falk Invariant of Shi and Linial Arrangements

It is an open question to give a combinatorial interpretation of the Falk invariant of a hyperplane arrangement, i.e. the third rank of successive quotients in the lower central series of the fundamental group of the arrangement. In this article, we give a combinatorial formula for this invariant in the case of hyperplane arrangements that are complete lift representation of certain gain graphs. As a corollary, we compute the Falk invariant for the cone of the braid, Shi, Linial and semiorder arrangements.

A Symmetric Function of Increasing Forests

Abstract For an indifference graph G, we define a symmetric function of increasing spanning forests of G. We prove that this symmetric function satisfies certain linear relations, which are also satisfied by the chromatic quasisymmetric function and unicellular $\textrm {LLT}$ polynomials. As a consequence, we give a combinatorial interpretation of the coefficients of the $\textrm {LLT}$ polynomial in the elementary basis (up to a factor of a power of $(q-1)$ ), strengthening the description given in [4].

Isodual and self-dual codes from graphs

Binary linear codes are constructed from graphs, in particular, by the generator matrix [In|A] where A is the adjacency matrix of a graph on n vertices. A combinatorial interpretation of the minimum distance of such codes is given. We also present graph theoretic conditions for such linear codes to be Type I and Type II self-dual. Several examples of binary linear codes produced by well-known graph classes are given.

Enumeration of weighted paths on a digraph and block hook determinant

Abstract In this article, we evaluate determinants of “block hook” matrices, which are block matrices consist of hook matrices. In particular, we deduce that the determinant of a block hook matrix factorizes nicely. In addition we give a combinatorial interpretation of the aforesaid factorization property by counting weighted paths in a suitable weighted digraph.

On the uniformity of the approximation for $r$-associated Stirling numbers of the second Kind

The $r$-associated Stirling numbers of the second kind are a natural extension of Stirling numbers of the second kind. A combinatorial interpretation of $r$-associated Stirling numbers of the second kind is the number of ways to partition $n$ elements into $m$ subsets such that each subset contains at least $r$ elements. Calculating the associated Stirling numbers is typically done with a recurrence relation or a generating function that are computationally expensive or alternatively with a closed-form that is practical for only a limited parameter range. In 1994 Hennecart proposed an approximation for the $r$-associated Stirling numbers that is fast to compute, is amenable to analysis over a wide range of parameters, and is conjectured to be asymptotically tight. There are a few other approximations for the associated Stirling numbers, but none of them are as general as Hennecart's. However, until this work, Hennecart's approximation had been utilized without a proper justification due to the absence of a rigorous proof. This work provides a proof of the uniformity of the Hennecart approximation.

On the uniformity of the approximation for $r$-associated Stirling numbers of the second Kind

The $r$-associated Stirling numbers of the second kind are a natural extension of Stirling numbers of the second kind. A combinatorial interpretation of $r$-associated Stirling numbers of the second kind is the number of ways to partition $n$ elements into $m$ subsets such that each subset contains at least $r$ elements. Calculating the associated Stirling numbers is typically done with a recurrence relation or a generating function that are computationally expensive or alternatively with a closed-form that is practical for only a limited parameter range. In 1994 Hennecart proposed an approximation for the $r$-associated Stirling numbers that is fast to compute, is amenable to analysis over a wide range of parameters, and is conjectured to be asymptotically tight. There are a few other approximations for the associated Stirling numbers, but none of them are as general as Hennecart's. However, until this work, Hennecart's approximation had been utilized without a proper justification due to the absence of a rigorous proof. This work provides a proof of the uniformity of the Hennecart approximation.

Combinatorics of two second-order mock theta functions

We introduce combinatorial interpretations of the coefficients of two second-order mock theta functions. Then, we provide a bijection that relates the two combinatorial interpretations for each function. By studying other special cases of the multivariate identity proved by the bijection, we obtain new combinatorial interpretations for the coefficients of Watson’s third-order mock theta function [Formula: see text] and Ramanujan’s third-order mock theta function [Formula: see text].

P.013 The neural signature of confidence abnormalities in compulsive disorders: obsessive-compulsive disorder and gambling disorder

We show that Γ-free fillings and lonesum fillings of Ferrers shapes are equinumerous by applying a previously defined bijection on matrices for this more general case and by constructing a new bijection between Callan sequences and Dumont-like permutations. As an application, we give a new combinatorial interpretation of Genocchi numbers in terms of Callan sequences.Further, we recover some of Hetyei’s results on alternating acyclic tournaments. Finally, we present an interesting result in the case of certain other special shapes.

Binary (k, k)-Designs

We introduce and investigate binary (k,k)-designs, a special case of T-designs. Our combinatorial interpretation relates (k,k)-designs to the binary orthogonal arrays. We derive a general linear programming bound and propose as a consequence a universal bound on the minimum possible cardinality of (k,k)-designs for fixed k and n. Designs which attain our bound are investigated.

CRYSTAL STRUCTURES FOR SYMMETRIC GROTHENDIECK POLYNOMIALS

The symmetric Grothendieck polynomials representing Schubert classes in the $K$-theory of Grassmannians are generating functions for semistandard set-valued tableaux. We construct a type $A_n$ crystal structure on these tableaux. This crystal yields a new combinatorial formula for decomposing symmetric Grothendieck polynomials into Schur polynomials. For single-columns and single-rows, we give a new combinatorial interpretation of Lascoux polynomials (K-analogs of Demazure characters) by constructing a K-theoretic analog of crystals with an appropriate analog of a Demazure crystal. We relate our crystal structure to combinatorial models using excited Young diagrams, Gelfand-Tsetlin patterns via the $5$-vertex model, and biwords via Hecke insertion to compute symmetric Grothendieck polynomials.

Simple Maps, Hurwitz Numbers, and Topological Recursion

We introduce the notion of fully simple maps, which are maps with non self-intersecting disjoint boundaries. In contrast, maps where such a restriction is not imposed are called ordinary. We study in detail the combinatorics of fully simple maps with topology of a disk or a cylinder. We show that the generating series of simple disks is given by the functional inversion of the generating series of ordinary disks. We also obtain an elegant formula for cylinders. These relations reproduce the relation between moments and (higher order) free cumulants established by Collins et al. [22], and implement the symplectic transformation x leftrightarrow y on the spectral curve in the context of topological recursion. We conjecture that the generating series of fully simple maps are computed by the topological recursion after exchange of x and y. We propose an argument to prove this statement conditionally to a mild version of the symplectic invariance for the 1-hermitian matrix model, which is believed to be true but has not been proved yet. Our conjecture can be considered as a combinatorial interpretation of the property of symplectic invariance of the topological recursion. Our argument relies on an (unconditional) matrix model interpretation of fully simple maps, via the formal hermitian matrix model with external field. We also deduce a universal relation between generating series of fully simple maps and of ordinary maps, which involves double monotone Hurwitz numbers. In particular, (ordinary) maps without internal faces—which are generated by the Gaussian Unitary Ensemble—and with boundary perimeters (lambda _1,ldots ,lambda _n) are strictly monotone double Hurwitz numbers with ramifications lambda above infty and (2,ldots ,2) above 0. Combining with a recent result of Dubrovin et al. [24], this implies an ELSV-like formula for these Hurwitz numbers.

Theta operators, refined Delta conjectures, and coinvariants

We introduce the family of Theta operators Θf indexed by symmetric functions f that allow us to conjecture a compositional refinement of the Delta conjecture of Haglund, Remmel and Wilson [23] for Δen−k−1′en. We show that the 4-variable Catalan theorem of Zabrocki [31] is precisely the Schröder case of our compositional Delta conjecture, and we show how to relate this conjecture to the Dyck path algebra introduced by Carlsson and Mellit in [6], extending one of their results.Again using the Theta operators, we conjecture a touching refinement of the generalized Delta conjecture for ΔhmΔen−k−1′en, and prove the case k=0, which was also conjectured in [23], extending the shuffle theorem of Carlsson and Mellit to a generalized shuffle theorem for Δhm∇en. Moreover we show how this implies the case k=0 of our generalized Delta square conjecture for [n−k]t[n]tΔhmΔen−kω(pn), extending the square theorem of Sergel [27] to a generalized square theorem for Δhm∇ω(pn).Still the Theta operators will provide a conjectural formula for the Frobenius characteristic of super-diagonal coinvariants with two sets of Grassmannian variables, extending the one of Zabrocki in [30] for the case with one set of such variables. We propose a combinatorial interpretation of this last formula at q=1, leaving open the problem of finding a dinv statistic that gives the whole symmetric function.

On the sum of parts with multiplicity at least 2 in all the partitions of n

In this paper, we investigate the sum of distinct parts that appear at least 2 times in all the partitions of [Formula: see text] providing new combinatorial interpretations for this sum. A connection with subsets of [Formula: see text] is given in this context. We provide two different proofs of our results: analytic and combinatorial. In addition, considering the multiplicity of parts in a partition, we provide a combinatorial proof of the truncated pentagonal number theorem of Andrews and Merca.

A meta-analysis of SARS-CoV-2 patients identifies the combinatorial significance of D-dimer, C-reactive protein, lymphocyte, and neutrophil values as a predictor of disease severity.

BackgroundSevere acute respiratory syndrome coronavirus 2 (SARS‐CoV‐2), known to be the causative agent of COVID‐19, has led to a worldwide pandemic. At presentation, individual clinical laboratory blood values, such as lymphocyte counts or C‐reactive protein (CRP) levels, may be abnormal and associated with disease severity. However, combinatorial interpretation of these laboratory blood values, in the context of COVID‐19, remains a challenge.MethodsTo assess the significance of multiple laboratory blood values in patients with SARS‐CoV‐2 and develop a COVID‐19 predictive equation, we conducted a literature search using PubMed to seek articles that included defined laboratory data points along with clinical disease progression. We identified 9846 papers, selecting primary studies with at least 20 patients for univariate analysis to identify clinical variables predicting nonsevere and severe COVID‐19 cases. Multiple regression analysis was performed on a training set of patient studies to generate severity predictor equations, and subsequently tested on a validation cohort of 151 patients who had a median duration of observation of 14 days.ResultsTwo COVID‐19 predictive equations were generated: one using four variables (CRP, D‐dimer levels, lymphocyte count, and neutrophil count), and another using three variables (CRP, lymphocyte count, and neutrophil count). In adult and pediatric populations, the predictive equations exhibited high specificity, sensitivity, positive predictive values, and negative predictive values.ConclusionUsing the generated equations, the outcomes of COVID‐19 patients can be predicted using commonly obtained clinical laboratory data. These predictive equations may inform future studies evaluating the long‐term follow‐up of COVID‐19 patients.

The Q-generating Function for Graphs with Application

For a simple connected graph G, the Q-generating function of the numbers $$N_k$$ of semi-edge walks of length k in G is defined by $$W_Q(t)=\sum \nolimits _{k = 0}^\infty {N_k t^k }$$ . This paper reveals that the Q-generating function $$W_Q(t)$$ may be expressed in terms of the Q-polynomials of the graph G and its complement $$\overline{G}$$ . Using this result, we study some Q-spectral properties of graphs and compute the Q-polynomials for some graphs obtained from various graph operations, such as the complement graph of a regular graph, the join of two graphs and the (edge)corona of two graphs. As another application of the Q-generating function $$W_Q(t)$$ , we also give a combinatorial interpretation of the Q-coronal of G, which is defined to be the sum of the entries of the matrix $$(\lambda I_n-Q(G))^{-1}$$ . This result may be used to obtain the many alternative calculations of the Q-polynomials of the (edge)corona of two graphs. Further, we also compute the Q-generating functions of the join of two graphs and the complete multipartite graphs.

Some multivariate polynomials for doubled permutations

<p style='text-indent:20px;'>Flajolet and Françon [European. J. Combin. 10 (1989) 235-241] gave a combinatorial interpretation for the Taylor coefficients of the Jacobian elliptic functions in terms of doubled permutations. We show that a multivariable counting of the doubled permutations has also an explicit continued fraction expansion generalizing the continued fraction expansions of Rogers and Stieltjes. The second goal of this paper is to study the expansion of the Taylor coefficients of the generalized Jacobian elliptic functions, which implies the symmetric and unimodal property of the Taylor coefficients of the generalized Jacobian elliptic functions. The main tools are the combinatorial theory of continued fractions due to Flajolet and bijections due to Françon-Viennot, Foata-Zeilberger and Clarke-Steingrímsson-Zeng.</p>

On q-Series and Split Lattice Paths

In this paper a natural question which arise to study the graphical aspect of split $$(n+t)$$ -color partitions, is answered by introducing a new class of lattice paths, called split lattice paths. A direct bijection between split $$(n+t)$$ -color partitions and split lattice paths is proved. This new combinatorial object is applied to give new combinatorial interpretations of two basic functions of Gordon-McIntosh. Some generalized q-series are also discussed. We further explore these paths by providing combinatorial interpretations of some Rogers-Ramanujan type identities which reveal their rich structure and great potential for further research.

On the roots of the Poupard and Kreweras polynomials

The Poupard polynomials are polynomials in one variable with integer coefficients, with some close relationship to Bernoulli and tangent numbers. They also have a combinatorial interpretation. We prove that every Poupard polynomial has all its roots on the unit circle. We also obtain the same property for another sequence of polynomials introduced by Kreweras and related to Genocchi numbers. This is obtained through a general statement about some linear operators acting on palindromic polynomials.

The $1/k$-Eulerian Polynomials of Type $B$

In this paper, we define the $1/k$-Eulerian polynomials of type $B$. Properties of these polynomials, including combinatorial interpretations, recurrence relations and $\gamma$-positivity are studied. In particular, we show that the $1/k$-Eulerian polynomials of type $B$ are $\gamma$-positive when $k>0$. Moreover, we define the $1/k$-derangement polynomials of type $B$, denoted $d_n^B(x;k)$. We show that the polynomials $d_n^B(x;k)$ are bi-$\gamma$-positive when $k\geq 1/2$. In particular, we get a symmetric decomposition of the polynomials $d_n^B(x;1/2)$ in terms of the classical derangement polynomials.