arXiv subjects

Toufik Mansour

Publications and source records attributed to Toufik Mansour.

At least 127 records · Page 7Linked to original sources

arXiv2007

Noncrossing normal ordering for functions of boson operators

Normally 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 few years, normally ordered forms have been shown to have a rich combinatorial structure, mainly in virtue of a link with the theory of partitions. In this paper, we attempt to enrich this link. By considering linear representations of noncrossing partitions, we define the notion of noncrossing normal ordering. Given the growing interest in noncrossing partitions, because of their many unexpected connections (like, for example, with free probability), noncrossing normal ordering appears to be an intriguing notion. We explicitly give the noncrossing normally ordered form of the functions (a^{r}(a^†)^{s})^{n}) and (a^{r}+(a^†)^{s})^{n}, plus various special cases. We are able to establish for the first time bijections between noncrossing contractions of these functions, k-ary trees and sets of lattice paths.

quant-ph↗

arXiv2007

On pattern-avoiding partitions

A \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 by a \emph{canonical sequence} $π_1,π_2,...c,π_n$, with $π_i=j$ if $i\in B_j$. We say that a partition $π$ \emph{contains} a partition $σ$ if the canonical sequence of $π$ contains a subsequence that is order-isomorphic to the canonical sequence of $σ$. Two partitions $σ$ and $σ'$ are \emph{equivalent}, if there is a size-preserving bijection between $σ$-avoiding and $σ'$-avoiding partitions. We determine several infinite families of sets of equivalent patterns; for instance, we prove that there is a bijection between $k$-noncrossing and $k$-nonnesting partitions, with a notion of crossing and nesting based on the canonical sequence. We also provide new combinatorial interpretations of the Catalan numbers and the Stirling numbers. Using a systematic computer search, we verify that our results characterize all the pairs of equivalent partitions of size at most seven. We also present a correspondence between set partitions and fillings of Ferrers shapes and stack polyominoes. This correspondence allows us to apply recent results on polyomino fillings in the study of partitions, and conversely, some of our results on partitions imply new results on polyomino fillings and ordered graphs.

math.CO↗

arXiv2007

Some families of density matrices for which separability is easily tested

We reconsider density matrices of graphs as defined in [quant-ph/0406165]. The density matrix of a graph is the combinatorial laplacian of the graph normalized to have unit trace. We describe a simple combinatorial condition (the "degree condition") to test separability of density matrices of graphs. The condition is directly related to the PPT-criterion. We prove that the degree condition is necessary for separability and we conjecture that it is also sufficient. We prove special cases of the conjecture involving nearest point graphs and perfect matchings. We observe that the degree condition appears to have value beyond density matrices of graphs. In fact, we point out that circulant density matrices and other matrices constructed from groups always satisfy the condition and indeed are separable with respect to any split. The paper isolates a number of problems and delineates further generalizations.

quant-ph↗

arXiv2007

On the normal ordering of multi-mode boson operators

In 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 problem of the single-mode case. To describe the normal ordering in the multi-mode case for each mode a colour is introduced and coloured contractions are considered. A depiction for coloured contractions via coloured linear representations is given. In analogy to the single-mode case associated coloured Stirling numbers are defined as coefficients appearing in the process of normal ordering powers of the number operators. Several properties of these coloured Stirling numbers are discussed.

quant-ph↗

arXiv2007

Wick's theorem for q-deformed boson operators

In 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'' the normally ordered form of a general expression in the creation and annihilation operators can be written as a sum over all q-weighted Feynman diagrams, representing Wick's theorem in the present context.

quant-ph↗

arXiv2007

Dyck paths with coloured ascents

We 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 cases enumeration gives new expression for sequences enumerating these structures.

math.CO↗

arXiv2006

A generalization of boson normal ordering

In 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 onto the combinatorics arising from algebraic and Fock space properties of boson operators.

quant-ph↗

arXiv2006

Excedance number for involutions in complex reflection groups

We 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 our results.

math.CO↗

arXiv2006

Restricted Dumont permutations, Dyck paths, and noncrossing partitions

We 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 of length 3 and 4 and give a natural bijection between 3142-avoiding Dumont permutations of the second kind and noncrossing partitions that uses cycle decomposition, as well as bijections between 132-, 231- and 321-avoiding Dumont permutations and Dyck paths. Finally, we enumerate Dumont permutations of the first kind simultaneously avoiding certain pairs of 4-letter patterns and another pattern of arbitrary length.

math.CO↗

arXiv2006

Restricted Motzkin permutations, Motzkin paths, continued fractions, and Chebyshev polynomials

We say that a permutation $π$ is a Motzkin permutation if it avoids 132 and there do not exist $a<b$ such that $π_a<π_b<π_{b+1}$. We study the distribution of several statistics in Motzkin permutations, including the length of the longest increasing and decreasing subsequences and the number of rises and descents. We also enumerate Motzkin permutations with additional restrictions, and study the distribution of occurrences of fairly general patterns in this class of permutations.

math.CO↗

arXiv2006

Partially ordered patterns and compositions

A 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 [Partially ordered generalized patterns, Discrete Math. 298 (2005), 212-229; Introduction to partially ordered patterns, Discrete Appl. Math., to appear], and studied in the set of $k$-ary words by Kitaev and Mansour [Partially ordered generalized patterns and $k$-ary words, Annals of Combinatorics 7 (2003) 191-200]. Moreover, Kitaev et al. [S. Kitaev, T. McAllister and K. Petersen, Enumerating segmented patterns in compositions and encoding with restricted permutations, preprint] introduced segmented POPs in compositions. In this paper, we study avoidance of POPs in compositions and generalize results for avoidance of POPs in permutations and words. Specifically, we obtain results for the generating functions for the number of compositions that avoid shuffle patterns and multi-patterns. In addition, we give the generating function for the distribution of the maximum number of non-overlapping occurrences of a segmented POP $τ$ (that is allowed to have repeated letters) among the compositions of $n$ with $m$ parts in a given set, provided we know the generating function for the number of compositions of $n$ with $m$ parts in the given set that avoid $τ$. This result is a $q$-analogue of the main result in [S. Kitaev, T. Mansour, Partially ordered generalized patterns and $k$-ary words, Annals of Combinatorics 7 (2003) 191-200].

math.CO↗

arXiv2006

Grid polygons from permutations and their enumeration by the kernel method

A 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 of permutations whose consecutive-minima polygons satisfy specific geometric conditions. We deal with 2-variate and 3-variate generating functions involving derivatives, cases which are not routinely solved by the kernel method.

math.CO↗

arXiv2005

On the X-rays of permutations

The 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 combinations of permutation matrices. This paper is an invitation to the study of X-rays of permutations from a combinatorial point of view. We present connections between these objects and nondecreasing differences of permutations, zero-sum arrays, decomposable permutations, score sequences of tournaments, queens' problems and rooks' problems.

math.CO↗

arXiv2005

Matchings Avoiding Partial Patterns

We 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 Schröder paths without peaks at level one and matchings avoiding both patterns 12312 and 121323. Such objects are counted by the super-Catalan numbers or the little Schröder numbers. A refinement of the super-Catalan numbers is obtained by fixing the number of crossings in the matchings. In the sense of Wilf-equivalence, we find that the patterns 12132, 12123, 12321, 12231, 12213 are equivalent to 12312.

math.CO↗

arXiv2004

Involutions Restricted by 3412, Continued Fractions, and Chebyshev Polynomials

We 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 involution avoid another pattern $τ$ or contain $τ$ exactly once. In many cases we express these generating functions in terms of Chebyshev polynomials of the second kind.

math.CO↗

arXiv2004

Counting occurrences of 3412 in an involution

We 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.

math.CO↗

arXiv2003

Independent sets in certain classes of (almost) regular graphs

We 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 the corresponding bijections.

math.CO↗

arXiv2003

Counting rises, levels, and drops in compositions

A 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 are the same in the given or in reverse order. In this paper we study the generating function for the number of compositions (respectively palindromic compositions) of $n$ with $m$ parts in a given set $A\subseteq\NN$ with respect to the number of rises, levels, and drops. As a consequence, we derive all the previously known results for this kind of problem, as well as many new results.

math.CO↗