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Generalizations of some identities involving the fibonacci numbersJan 15 2003In this paper we study the sum $$\sum_{j_1+j_2+...+j_d=n}\prod_{i=1}^d F_{k\cdot j_i},$$ where $d\geq2$ and $k\geq1$.

$q$-deformed conformable fractional Natural transformNov 06 2018In this paper, we develop a new deformation and generalization of the Natural integral transform based on the conformable fractional $q$-derivative. We obtain transformation of some deformed functions and apply the transform for solving linear differential ... More

Modelling x-ray tomography using integer compositionsAug 12 2015The x-ray process is modelled using integer compositions as a two dimensional analogue of the object being x-rayed, where the examining rays are modelled by diagonal lines with equation $x-y=n$ for non negative integers $n$. This process is essentially ... More

Some recursive formulas for Selberg-type integralsDec 17 2009A set of recursive relations satisfied by Selberg-type integrals involving monomial symmetric polynomials are derived, generalizing previously known results. These formulas provide a well-defined algorithm for computing Selberg-Schur integrals whenever ... More

A monotonicity property for generalized Fibonacci sequencesOct 25 2014Given k>1, let a_n be the sequence defined by the recurrence a_n=c_1a_{n-1}+c_2a_{n-2}+...+c_ka_{n-k} for n>=k, with initial values a_0=a_1=...=a_{k-2}=0 and a_{k-1}= 1. We show under a couple of assumptions concerning the constants c_i that the ratio ... More

New degenerated polynomials arising from non-classical Umbral CalculusNov 06 2018We introduce new generalizations of the Bernoulli, Euler, and Genocchi polynomials and numbers based on the Carlitz-Tsallis degenerate exponential function and concepts of the Umbral Calculus associated with it. Also, we present generalizations of some ... More

The 1/k-Eulerian polynomials and k-Stirling permutationsSep 23 2014In this paper, we establish a connection between the 1/k-Eulerian polynomials introduced by Savage and Viswanathan (Electron. J. Combin. 19(2012), P9) and k-Stirling permutations. We also introduce the dual set of Stirling permutations.

Wick's theorem for q-deformed boson operatorsMar 11 2007In this paper combinatorial aspects of normal ordering arbitrary words in the creation and annihilation operators of the q-deformed boson are discussed. In particular, it is shown how by introducing appropriate q-weights for the associated ``Feynman diagrams'' ... More

Wilf classification of triples of 4-letter patternsMay 16 2016We determine all 242 Wilf classes of triples of 4-letter patterns by showing that there are 32 non-singleton Wilf classes. There are 317 symmetry classes of triples of 4-letter patterns and after computer calculation of initial terms, the problem reduces ... More

Motzkin numbers of higher rank: Generating function and explicit expressionApr 21 2007May 13 2007The generating function and an explicit expression is derived for the (colored) Motzkin numbers of higher rank introduced recently. Considering the special case of rank one yields the corresponding results for the conventional colored Motzkin numbers ... More

A characterization of horizontal visibility graphs and combinatorics on wordsOct 09 2010An Horizontal Visibility Graph (for short, HVG) is defined in association with an ordered set of non-negative reals. HVGs realize a methodology in the analysis of time series, their degree distribution being a good discriminator between randomness and ... More

Diffusion on an Ising chain with kinksJun 30 2008Jul 28 2009We count the number of histories between the two degenerate minimum energy configurations of the Ising model on a chain, as a function of the length n and the number d of kinks that appear above the critical temperature. This is equivalent to count permutations ... More

Apostol-Euler polynomials arising from umbral calculusFeb 13 2013In this paper, by using the orthogonality type as defined in the umbral calculus, we derive explicit formula for several well known polynomials as a linear combination of the Apostol-Euler polynomials.

Restricted Stirling permutationsJul 20 2016In this paper, we study the generating functions for the number of pattern restricted Stirling permutations with a given number of plateaus, descents and ascents. Properties of the generating functions, including symmetric properties and explicit formulas ... More

Recurrence relations in counting the pattern 13-2 in flattened permutationsOct 15 2014We prove that the generating function for the number of flattened permutations having a given number of occurrences of the pattern 13-2 is rational, by using the recurrence relations and the kernel method.

Counting occurrences of 3412 in an involutionJan 18 2004We study the generating function for the number of involutions on $n$ letters containing exactly $r\gs0$ occurrences of 3412. It is shown that finding this function for a given $r$ amounts to a routine check of all involutions on $2r+1$ letters.

On moments of the integrated exponential Brownian motionSep 20 2015Jun 24 2016We present new exact expressions for a class of moments for the geometric Brownian motion, in terms of determinants, obtained using a recurrence relation and combinatorial arguments for the case of a Ito's Wiener process. We then apply the obtained exact ... More

A note on sum of k-th power of Horadam's sequenceFeb 02 2003Let $w_{n+2}=pw_{n+1}+qw_{n}$ for $n\geq0$ with $w_0=a$ and $w_1=b$. In this paper we find an explicit expression, in terms of determinants, for $\sum_{n\geq0} w_n^kx^n$ for any $k\geq1$. As a consequence, we derive all the previously known results for ... More

Squaring the terms of an $\ell^{th}$ order linear recurrenceMar 12 2003We find an explicit formula for the generating function for the squaring the terms of an $\ell^{th}$ order linear recurrence.

Restricted even permutations and Chebyshev polynomialsFeb 02 2003We study generating functions for the number of even (odd) permutations on n letters avoiding 132 and an arbitrary permutation $\tau$ on k letters, or containing $\tau$ exactly once. In several interesting cases the generating function depends only on ... More

On the Complementary Equienergetic GraphsJul 30 2019The energy of a simple graph $G$, denoted by $\mathcal{E}(G)$, is the sum of the absolute values of the eigenvalues of $G$. Two $n$-vertex graphs with the same energies are called equienergetic graphs. A graph $G$ with the property $G\cong \overline{G}$ ... More

Recursions for Excedance number in some permutations groupsFeb 15 2007Jun 03 2008The excedance number for S_n is known to have an Eulerian distribution. Nevertheless, the classical proof uses descents rather than excedances. We present a direct recursive proof which seems to be folklore and extend it to the colored permutation groups ... More

Bilinear Forms on Skein Modules and Steps in Dyck PathsNov 03 2010Jan 13 2011We use Jones-Wenzl idempotents to construct bases for the relative Kauffman bracket skein module of a square with n points colored 1 and one point colored h. We consider a natural bilinear form on this skein module. We calculate the determinant of the ... More

Counting occurrences of a pattern of type (1,2) or (2,1) in permutationsOct 03 2001Babson and Steingr\'{\i}msson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Claesson presented a complete solution for the number of permutations avoiding ... More

Enumerating permutations avoiding a pair of Babson-Steingrimsson patternsJul 06 2001Mar 24 2010Babson and Steingr\`imsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Subsequently, Claesson presented a complete solution for the number of permutations ... More

On pattern-avoiding partitionsMar 29 2007A \emph{set partition} of the set $[n]=\{1,...c,n\}$ is a collection of disjoint blocks $B_1,B_2,...c, B_d$ whose union is $[n]$. We choose the ordering of the blocks so that they satisfy $\min B_1<\min B_2<...b<\min B_d$. We represent such a set partition ... More

Enumerations of bargraphs with respect to corner statisticsAug 05 2018We study the enumeration of bargraphs with respect to some corner statistics. We find generating functions for the number of bargraphs that tracks the corner statistics of interest, the number of cells, and the number of columns. The bargraph representation ... More

Refined Restricted Permutations Avoiding Subsets of Patterns of Length ThreeMar 30 2002Define $S_n^k(T)$ to be the set of permutations of $\{1,2,...,n\}$ with exactly $k$ fixed points which avoid all patterns in $T \subseteq S_m$. We enumerate $S_n^k(T)$, $T \subseteq S_3$, for all $|T| \geq 2$ and $0 \leq k \leq n$.

On the number of combinations without certain separationsMay 09 2008In this paper we enumerate the number of ways of selecting $k$ objects from $n$ objects arrayed in a line such that no two selected ones are separated by $m-1,2m-1,...,pm-1$ objects and provide three different formulas when $m,p\geq 1$ and $n\geq pm(k-1)$. ... More

Avoiding maximal parabolic subgroups of S_kJun 21 2000We find an explicit expression for the generating function of the number of permutations in S_n avoiding a subgroup of S_k generated by all but one simple transpositions. The generating function turns out to be rational, and its denominator is a rook ... More

On the normal ordering of multi-mode boson operatorsJan 25 2007Mar 19 2007In this article combinatorial aspects of normal ordering annihilation and creation operators of a multi-mode boson system are discussed. The modes are assumed to be coupled since otherwise the problem of normal ordering is reduced to the corresponding ... More

Permutations avoiding 312 and another pattern, Chebyshev polynomials and longest increasing subsequencesAug 16 2018We study the longest increasing subsequence problem for random permutations from $S_n(312,\tau)$, the set of all permutations of length $n$ avoiding the pattern $312$ and another pattern $\tau$, under the uniform probability distribution. We determine ... More

Enumeration of $(k,2)$-noncrossing partitionsAug 08 2008A set partition is said to be $(k,d)$-noncrossing if it avoids the pattern $12... k12... d$. We find an explicit formula for the ordinary generating function of the number of $(k,d)$-noncrossing partitions of $\{1,2,...,n\}$ when $d=1,2$.

Words restricted by patterns with at most 2 distinct lettersOct 04 2001We find generating functions for the number of words avoiding certain patterns or sets of patterns on at most 2 distinct letters and determine which of them are equally avoided. We also find the exact number of words avoiding certain patterns and provide ... More

Excedance numbers for permutations in complex reflection groupsApr 23 2007Recently, Bagno, Garber and Mansour studied a kind of excedance number on the complex reflection groups and computed its multidistribution with the number of fixed points on the set of involutions in these groups. In this note, we consider the similar ... More

Chebyshev Polynomials and Statistics on a New Collection of Words in the Catalan FamilyJul 14 2014Recently, a new class of words, denoted by L_n, was shown to be in bijection with a subset of the Dyck paths of length 2n having cardinality given by the (n-1)-st Catalan number. Here, we consider statistics on L_n recording the number of occurrences ... More

Involutions Restricted by 3412, Continued Fractions, and Chebyshev PolynomialsJan 18 2004We study generating functions for the number of involutions, even involutions, and odd involutions in $S_n$ subject to two restrictions. One restriction is that the involution avoid 3412 or contain 3412 exactly once. The other restriction is that the ... More

On Linear Differential Equations Involving a Para-Grassmann VariableJul 15 2009As a first step towards a theory of differential equations involving para-Grassmann variables the linear equations with constant coefficients are discussed and solutions for equations of low order are given explicitly. A connection to n-generalized Fibonacci ... More

Finite automata and pattern avoidance in wordsSep 17 2003We say that a word $w$ on a totally ordered alphabet avoids the word $v$ if there are no subsequences in $w$ order-equivalent to $v$. In this paper we suggest a new approach to the enumeration of words on at most $k$ letters avoiding a given pattern. ... More

Bell Polynomials and $k$-generalized Dyck PathsMay 09 2008A {\em k-generalized Dyck path} of length $n$ is a lattice path from $(0,0)$ to $(n,0)$ in the plane integer lattice $\mathbb{Z}\times\mathbb{Z}$ consisting of horizontal-steps $(k, 0)$ for a given integer $k\geq 0$, up-steps $(1,1)$, and down-steps $(1,-1)$, ... More

Counting rises, levels, and drops in compositionsOct 14 2003A composition of $n\in\NN$ is an ordered collection of one or more positive integers whose sum is $n$. The number of summands is called the number of parts of the composition. A palindromic composition of $n$ is a composition of $n$ in which the summands ... More

Five subsets of permutations enumerated as weak sorting permutationsFeb 16 2016We show that the number of members of S_n avoiding any one of five specific triples of 4-letter patterns is given by sequence A111279 in OEIS, which is known to count weak sorting permutations. By numerical evidence, there are no other (non-trivial) triples ... More

$q$-Bernstein functions and applicationsFeb 09 2016We characterize of the $q$-Bernstein functions in terms of $q$-Laplace transform. Moreover, we present several results of $q$-completely monotonic, $q$-log completely monotonic and $q$-Bernstein functions.

Counting paths in Bratteli diagrams for SU(2)_kJun 30 2008Mar 20 2009It is known that the Hilbert space dimensionality for quasiparticles in an SU(2)_k Chern-Simons-Witten theory is given by the number of directed paths in certain Bratteli diagrams. We present an explicit formula for these numbers for arbitrary k. This ... More

On avoidance of patterns of the form σ-τ by words over a finite alphabetMar 10 2014Vincular or dashed patterns resemble classical patterns except that some of the letters within an occurrence are required to be adjacent. We prove several infinite families of Wilf-equivalences for k-ary words involving vincular patterns containing a ... More

Grid polygons from permutations and their enumeration by the kernel methodMar 09 2006A grid polygon is a polygon whose vertices are points of a grid. We define an injective map between permutations of length n and a subset of grid polygons on n vertices, which we call consecutive-minima polygons. By the kernel method, we enumerate sets ... More

Counting occurences of 132 in a permutationMay 09 2001Aug 02 2001We study the generating function for the number of permutations on n letters containing exactly $r\gs0$ occurences of 132. It is shown that finding this function for a given r amounts to a routine check of all permutations in $S_{2r}$.

Words restricted by 3-letter generalized multipermutation patternsDec 27 2001We find exact formulas and/or generating functions for the number of words avoiding 3-letter generalized multipermutation patterns and find which of them are equally avoided.

Some enumerative results related to ascent sequencesJul 16 2012An ascent sequence is one consisting of non-negative integers in which the size of each letter is restricted by the number of ascents preceding it in the sequence. Ascent sequences have recently been shown to be related to (2+2)-free posets and a variety ... More

Separable d-permutations and guillotine partitionsMar 24 2008We characterize separable multidimensional permutations in terms of forbidden patterns and enumerate them by means of generating function, recursive formula and explicit formula. We find a connection between multidimensional permutations and guillotine ... More

Enumeration of small Wilf classes avoiding 1324 and two other 4-letter patternsMay 02 2017Nov 12 2017Recently, it has been determined that there are 242 Wilf classes of triples of 4-letter permutation patterns by showing that there are 32 non-singleton Wilf classes. Moreover, the generating function for each triple lying in a non-singleton Wilf class ... More

Generalized q-Calkin-Wilf trees and c-hyper m-expansions of integersMar 13 2015A hyperbinary expansion of a positive integer n is a partition of n into powers of 2 in which each part appears at most twice. In this paper, we consider a generalization of this concept and a certain statistic on the corresponding set of expansions of ... More

Identities involving Narayana polynomials and Catalan numbersMay 09 2008We first establish the result that the Narayana polynomials can be represented as the integrals of the Legendre polynomials. Then we represent the Catalan numbers in terms of the Narayana polynomials by three different identities. We give three different ... More

Dyck paths with coloured ascentsJan 25 2007We introduce a notion of Dyck paths with coloured ascents. For several ways of colouring, we establish bijections between sets of such paths and other combinatorial structures, such as non-crossing trees, dissections of a convex polygon, etc. In some ... More

Restricted Motzkin permutations, Motzkin paths, continued fractions, and Chebyshev polynomialsOct 06 2006We say that a permutation $\pi$ is a Motzkin permutation if it avoids 132 and there do not exist $a<b$ such that $\pi_a<\pi_b<\pi_{b+1}$. We study the distribution of several statistics in Motzkin permutations, including the length of the longest increasing ... More

Combinatorial Gray codes for classes of pattern avoiding permutationsApr 16 2007Jan 09 2008The past decade has seen a flurry of research into pattern avoiding permutations but little of it is concerned with their exhaustive generation. Many applications call for exhaustive generation of permutations subject to various constraints or imposing ... More

Evaluation of spherical GJMS determinantsJul 23 2014An expression in the form of an easily computed integral is given for the determinant of the scalar GJMS operator on an odd--dimensional sphere. Manipulation yields a sum formula for the logdet in terms of the logdets of the ordinary conformal Laplacian ... More

231-Avoiding Involutions and Fibonacci NumbersSep 19 2002We use combinatorial and generating function techniques to enumerate various sets of involutions which avoid 231 or contain 231 exactly once. Interestingly, many of these enumerations can be given in terms of $k$-generalized Fibonacci numbers.

Permutations Which Avoid 1243 and 2143, Continued Fractions, and Chebyshev PolynomialsAug 06 2002Several authors have examined connections between permutations which avoid 132, continued fractions, and Chebyshev polynomials of the second kind. In this paper we prove analogues of some of these results for permutations which avoid 1243 and 2143. Using ... More

A Digital Binomial Theorem for Sheffer SequencesOct 29 2015We extend the digital binomial theorem to Sheffer polynomial sequences by demonstrating that their corresponding Sierpi\'nski matrices satisfy a multiplication property that is equivalent to the convolution identity for Sheffer sequences.

132-avoiding Two-stack Sortable Permutations, Fibonacci Numbers, and Pell NumbersMay 19 2002In 1990 West conjectured that there are $2(3n)!/((n+1)!(2n+1)!)$ two-stack sortable permutations on $n$ letters. This conjecture was proved analytically by Zeilberger in 1992. Later, Dulucq, Gire, and Guibert gave a combinatorial proof of this conjecture. ... More

Restricted Permutations, Fibonacci Numbers, and k-generalized Fibonacci NumbersMar 21 2002A permutation $\pi \in S_n$ is said to {\it avoid} a permutation $\sigma \in S_k$ whenever $\pi$ contains no subsequence with all of the same pairwise comparisons as $\sigma$. For any set $R$ of permutations, we write $S_n(R)$ to denote the set of permutations ... More

A $q$-Digital Binomial TheoremJun 26 2015We present a multivariable generalization of the digital binomial theorem from which a q-analog is derived as a special case.

Restricted Dumont permutations, Dyck paths, and noncrossing partitionsOct 06 2006We complete the enumeration of Dumont permutations of the second kind avoiding a pattern of length 4 which is itself a Dumont permutation of the second kind. We also consider some combinatorial statistics on Dumont permutations avoiding certain patterns ... More

Counting triangulations of some classes of subdivided convex polygonsApr 11 2016We compute the number of triangulations of a convex $k$-gon each of whose sides is subdivided by $r-1$ points. We find explicit formulas and generating functions, and we determine the asymptotic behaviour of these numbers as $k$ and/or $r$ tend to infinity. ... More

Finite automata, probabilistic method, and occurrence enumeration of a pattern in words and permutationsMay 14 2019The main theme of this paper is the enumeration of the occurrence of a pattern in words and permutations. We mainly focus on asymptotic properties of the sequence $f_r^v(k,n),$ the number of $n$-array $k$-ary words that contain a given pattern $v$ exactly ... More

A comment on Ryser's conjecture for intersecting hypergraphsSep 20 2007Let $\tau(\mathcal{H})$ be the cover number and $\nu(\mathcal{H})$ be the matching number of a hypergraph $\mathcal{H}$. Ryser conjectured that every $r$-partite hypergraph $\mathcal{H}$ satisfies the inequality $\tau(\mathcal{H}) \leq (r-1) \nu (\mathcal{H})$. ... More

Staircase patterns in words: subsequences, subwords, and separation numberAug 02 2019We revisit staircases for words and prove several exact as well as asymptotic results for longest left-most staircase subsequences and subwords and staircase separation number, the latter being defined as the number of consecutive maximal staircase subwords ... More

On ballistic deposition process on a stripMar 29 2019Jun 19 2019We revisit the model of the ballistic deposition studied in \cite{bdeposition} and prove several combinatorial properties of the random tree structure formed by the underlying stochastic process. Our results include limit theorems for the number of roots ... More

Noncrossing normal ordering for functions of boson operatorsJul 11 2006Apr 03 2007Normally ordered forms of functions of boson operators are important in many contexts in particular concerning Quantum Field Theory and Quantum Optics. Beginning with the seminal work of Katriel [Lett. Nuovo Cimento, 10(13):565--567, 1974], in the last ... More

Partial transpose of permutation matricesSep 21 2007Mar 22 2008The partial transpose of a block matrix M is the matrix obtained by transposing the blocks of M independently. We approach the notion of partial transpose from a combinatorial point of view. In this perspective, we solve some basic enumeration problems ... More

On ballistic deposition process on a stripMar 29 2019We revisit the model of the ballistic deposition studied in \cite{bdeposition} and prove several combinatorial properties of the random tree structure formed by the underlying stochastic process. Our results include limit theorems for the number of roots ... More

On Multiple Pattern Avoiding Set PartitionsJan 28 2013Jan 29 2013We study classes of set partitions determined by the avoidance of multiple patterns, applying a natural notion of partition containment that has been introduced by Sagan. We say that two sets S and T of patterns are equivalent if for each n, the number ... More

Partially ordered patterns and compositionsOct 01 2006A partially ordered (generalized) pattern (POP) is a generalized pattern some of whose letters are incomparable, an extension of generalized permutation patterns introduced by Babson and Steingrimsson. POPs were introduced in the symmetric group by Kitaev ... More

On the group of alternating colored permutationsJan 22 2014The group of alternating colored permutations is the natural analogue of the classical alternating group, inside the wreath product $\mathbb{Z}_r \wr S_n$. We present a 'Coxeter-like' presentation for this group and compute the length function with respect ... More

A generalization of boson normal orderingAug 09 2006Dec 12 2006In this paper we define generalizations of boson normal ordering. These are based on the number of contractions whose vertices are next to each other in the linear representation of the boson operator function. Our main motivation is to shed further light ... More

Restricted ascent sequences and Catalan numbersMar 27 2014Ascent sequences are those consisting of non-negative integers in which the size of each letter is restricted by the number of ascents preceding it and have been shown to be equinumerous with the (2+2)-free posets of the same size. Furthermore, connections ... More

Congruence successions in compositionsJul 28 2013A \emph{composition} is a sequence of positive integers, called \emph{parts}, having a fixed sum. By an \emph{$m$-congruence succession}, we will mean a pair of adjacent parts $x$ and $y$ within a composition such that $x\equiv y(\text{mod} m)$. Here, ... More

Excedance number for involutions in complex reflection groupsDec 07 2006We define the excedance number on the complex reflection groups and compute its multidistribution with the number of fixed points on the set of involutions in these groups. We use some recurrence formulas and generating functions manipulations to obtain ... More

Passing through a stack $k$ timesApr 13 2017Jul 02 2018We consider the number of passes a permutation needs to take through a stack if we only pop the appropriate output values and start over with the remaining entries in their original order. We define a permutation $\pi$ to be $k$-pass sortable if $\pi$ ... More

Counting descents, rises, and levels, with prescribed first element, in wordsDec 31 2006May 30 2007Recently, Kitaev and Remmel [Classifying descents according to parity, Annals of Combinatorics, to appear 2007] refined the well-known permutation statistic ``descent'' by fixing parity of one of the descent's numbers. Results in that paper were extended ... More

On the degeneracy of $SU(3)_k$ topological phasesSep 01 2010The ground state degeneracy of an $SU(N)_k$ topological phase with $n$ quasiparticle excitations is relevant quantity for quantum computation, condensed matter physics, and knot theory. It is an open question to find a closed formula for this degeneracy ... More

Height of records in partitions of a setAug 02 2019We study the restricted growth function associated with set partitions, and obtain exact formulas for the number of strong records with height one, the total of record heights over set of partitions, and the number of partitions with a given maximal height ... More

Independent sets in certain classes of (almost) regular graphsOct 23 2003We enumerate the independent sets of several classes of regular and almost regular graphs and compute the corresponding generating functions. We also note the relations between these graphs and other combinatorial objects and, in some cases, construct ... More

Normal ordering problem and the extensions of the Stirling grammarAug 01 2013The purpose of this paper is to investigate the connection between context-free grammars and normal ordering problem, and then to explore various extensions of the Stirling grammar. We present grammatical characterizations of several well known combinatorial ... More

Some combinatorial arrays related to the Lotka-Volterra systemApr 02 2014The purpose of this paper is to investigate the connection between the Lotka-Volterra system and combinatorics. We study several context-free grammars associated with the Lotka-Volterra system. Some combinatorial arrays, involving the Stirling numbers ... More

Counting descent pairs with prescribed colors in the colored permutation groupsSep 17 2007We define new statistics, (c, d)-descents, on the colored permutation groups Z_r \wr S_n and compute the distribution of these statistics on the elements in these groups. We use some combinatorial approaches, recurrences, and generating functions manipulations ... More

Passing through a stack $k$ times with reversalsAug 13 2018We consider a stack sorting algorithm where only the appropriate output values are popped from the stack and then any remaining entries in the stack are run through the stack in reverse order. We identify the basis for the $2$-reverse pass sortable permutations ... More

A notion of graph likelihood and an infinite monkey theoremApr 12 2013We play with a graph-theoretic analogue of the folklore infinite monkey theorem. We define a notion of graph likelihood as the probability that a given graph is constructed by a monkey in a number of time steps equal to the number of vertices. We present ... More

Nonlinear differential equation for Korobov numbersApr 15 2016In this paper, we present nonlinear differential equations for the generating functions for the Korobov numbers and for the Frobenuius-Euler numbers. As an application, we find an explicit expression for the nth derivative of 1/ log(1 + t).

The descent statistic on signed simsun permutationsMay 09 2016May 17 2016In this paper we study the generating polynomials obtained by enumerating signed simsun permutations by number of the descents. Properties of the polynomials, including the recurrence relations and generating functions are studied.

Recurrence relations for patterns of type $(2,1)$ in flattened permutationsJun 14 2013We consider the problem of counting the occurrences of patterns of the form $xy-z$ within flattened permutations of a given length. Using symmetric functions, we find recurrence relations satisfied by the distributions on $\mathcal{S}_n$ for the patterns ... More

Counting subwords in flattened permutationsJul 13 2013In this paper, we consider the number of occurrences of descents, ascents, 123-subwords, 321-subwords, peaks and valleys in flattened permutations, which were recently introduced by Callan in his study of finite set partitions. For descents and ascents, ... More

Combinatorics of Dumont differential system on the Jacobi elliptic functionsMar 02 2014In this paper, we relate Jacobi elliptic functions to several combinatorial structures, including the longest alternating subsequences, alternating runs and descents. The Dumont differential system on the Jacobi elliptic functions is defined by $D(x)=yz,~D(y)=xz,~D(z)=xy$. ... More

Smooth words and Chebyshev polynomialsSep 03 2008A word $\sigma=\sigma_1...\sigma_n$ over the alphabet $[k]=\{1,2,...,k\}$ is said to be {\em smooth} if there are no two adjacent letters with difference greater than 1. A word $\sigma$ is said to be {\em smooth cyclic} if it is a smooth word and in addition ... More

On the X-rays of permutationsJun 16 2005The X-ray of a permutation is defined as the sequence of antidiagonal sums in the associated permutation matrix. X-rays of permutation are interesting in the context of Discrete Tomography since many types of integral matrices can be written as linear ... More

New equivalences for pattern avoiding involutionsAug 10 2007Jan 22 2008We complete the Wilf classification of signed patterns of length 5 for both signed permutations and signed involutions. New general equivalences of patterns are given which prove Jaggard's conjectures concerning involutions in the symmetric group avoiding ... More

Matchings Avoiding Partial PatternsApr 17 2005We show that matchings avoiding certain partial patterns are counted by the 3-Catalan numbers. We give a characterization of 12312-avoiding matchings in terms of restrictions on the corresponding oscillating tableaux. We also find a bijection between ... More

Orion Routing Protocol for Delay-Tolerant NetworksMay 10 2012In this paper, we address the problem of efficient routing in delay tolerant network. We propose a new routing protocol dubbed as ORION. In ORION, only a single copy of a data packet is kept in the network and transmitted, contact by contact, towards ... More

Family of Subharmonic Functions and Separately Subharmonic FunctionsMay 31 2016Jul 31 2016We prove that a separately subharmonic function is subharmonic outside a closed set whose projections are closed nowhere dense with no bounded components. It generalizes a result due to U. Cegerell and A. Sadullaev. Then, given such a set, we construct ... More