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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., Comment: 10 pages, 1 figure

The $k$-arrangements are permutations whose fixed points are $k$-colored. We prove enumerative results related to statistics and patterns on $k$-arrangements, confirming several conjectures by Blitvi\'c and Steingr\'imsson. In particular, one of their conjectures regarding the equdistribution of the number of descents over the derangement form and the permutation form of $k$-arrangements is strengthened in two interesting ways. Moreover, as one application of the so-called Decrease Value Theorem, we calculate the generating function for a symmetric pair of Eulerian statistics over permutations arising in our study., Comment: 25 pages, 1 figure and 1 table

We prove a number of conjectures [arXiv:2005.04066] recently stated by P. Barry, related to the paperfolding sequence and the Rueppel sequence., this supersedes also the paper arXiv:2006.04708

Plane partitions have been widely studied in Mathematics since MacMahon. See, for example, the works by Andrews, Macdonald, Stanley, Sagan and Krattenthaler. The Schur process approach, introduced by Okounkov and Reshetikhin, and further developed by Borodin, Corwin, Corteel, Savelief and Vuleti\'c, has been proved to be a powerful tool in the study of various kinds of plane partitions. The exact enumerations of ordinary plane partitions, shifted plane partitions and cylindric partitions could be derived from two summation formulas for Schur processes, namely, the open summation formula and the cylindric summation formula. In this paper, we establish a new summation formula for Schur processes, called the complete summation formula. As an application, we obtain the generating function and the asymptotic formula for the number of doubled shifted plane partitions, which can be viewed as plane partitions `shifted at the two sides'. We prove that the order of the asymptotic formula depends only on the diagonal width of the doubled shifted plane partition, not on the profile (the skew zone) itself. By using the same methods, the generating function and the asymptotic formula for the number of symmetric cylindric partitions are also derived., Comment: 19 pages

We introduce the difference operator for functions defined on strict partitions and prove a polynomiality property for a summation involving the hook length and content statistics. As an application, several new hook-content formulas for strict partitions are derived., 16 pages

International audience; We prove several conjectures made by Z.-W. Sun on the existence of permutations conditioned by certain rational functions. Furthermore, we fully characterize all integer values of the "inverse difference" rational function. Our proofs consist of both investigation of the mathematical properties of the rational functions and brute-force attack by computer for finding special permutations.

In 1998, Allouche, Peyri\`{e}re, Wen and Wen showed that the Hankel determinant $H_n$ of the Thue-Morse sequence over $\{-1,1\}$ satisfies $H_n/2^{n-1}\equiv 1~(\mathrm{mod}~2)$ for all $n\geq 1$. Inspired by this result, Fu and Han introduced \emph{apwenian} sequences over $\{-1,1\}$, namely, $\pm 1$ sequences whose Hankel determinants satisfy $H_n/2^{n-1}\equiv 1~(\mathrm{mod}~2)$ for all $n\geq 1$, and proved with computer assistance that a few sequences are apwenian. In this paper, we obtain an easy to check criterion for apwenian sequences, which allows us to determine all apwenian sequences that are fixed points of substitutions of constant length. Let $f(z)$ be the generating functions of such apwenian sequences. We show that for all integer $b\ge 2$ with $f(1/b)\neq 0$, the real number $f(1/b)$ is transcendental and its irrationality exponent is equal to $2$. Besides, we also derive a criterion for zero-one apwenian sequences whose Hankel determinants satisfy $H_n\equiv 1~(\mathrm{mod}~2)$ for all $n\geq 1$. We find that the only zero-one apwenian sequence, among all fixed points of substitutions of constant length, is the period-doubling sequence. Various examples of apwenian sequences given by substitutions with projection are also given. Furthermore, we prove that all Sturmian sequences over $\{-1,1\}$ or $\{0,1\}$ are not apwenian. And we conjecture that fixed points of substitution of non-constant length over $\{-1,1\}$ or $\{0,1\}$ can not be apwenian., Comment: 27 pages

International audience; Inspired by the works of Dewar, Murty and Kotěšovec, we establish some useful theorems for asymptotic formulas. As an application, we obtain asymptotic formulas for the numbers of skew plane partitions and cylin-dric partitions. We prove that the order of the asymptotic formula for the skew plane partitions of fixed width depends only on the width of the region, not on the profile (the skew zone) itself, while this is not true for cylindric partitions.

The Euler numbers occur in the Taylor expansion of tan ⁡ ( x ) + sec ⁡ ( x ) \tan (x)+\sec (x) . Since Stieltjes, continued fractions and Hankel determinants of the even Euler numbers, on the one hand, of the odd Euler numbers, on the other hand, have been widely studied separately. However, no Hankel determinants of the (mixed) Euler numbers have been obtained. The reason for that is that some Hankel determinants of the Euler numbers are null. This implies that the Jacobi continued fraction of the Euler numbers does not exist. In the present paper, this obstacle is bypassed by using the Hankel continued fraction, instead of the J J -fraction. Consequently, an explicit formula for the Hankel determinants of the Euler numbers is being derived, as well as a full list of Hankel continued fractions and Hankel determinants involving Euler numbers. Finally, a new q q -analog of the Euler numbers E n ( q ) E_n(q) based on our continued fraction is proposed. We obtain an explicit formula for E n ( − 1 ) E_n(-1) and prove a conjecture by R. J. Mathar on these numbers.

The concept of $t$-difference operator for functions of partitions is introduced to prove a generalization of Stanley's theorem on polynomiality of Plancherel averages of symmetric functions related to contents and hook lengths. Our extension uses a generalization of the notion of Plancherel measure, based on walks in the Young lattice with steps given by the addition of $t$-hooks. It is well-known that the hook lengths of multiples of $t$ can be characterized by the Littlewood decomposition. Our study gives some further information on the contents and hook lengths of other congruence classes modulo $t$., Comment: 24 pages

The irrationality exponent of an irrational number $\xi$, which measures the approximation rate of $\xi$ by rationals, is in general extremely difficult to compute explicitly, unless we know the continued fraction expansion of $\xi$. Results obtained so far are rather fragmentary, and often treated case by case. In this work, we shall unify all the known results on the subject by showing that the irrationality exponents of large classes of automatic numbers and Mahler numbers (which are transcendental) are exactly equal to $2$. Our classes contain the Thue--Morse--Mahler numbers, the sum of the reciprocals of the Fermat numbers, the regular paperfolding numbers, which have been previously considered respectively by Bugeaud, Coons, and Guo, Wu and Wen, but also new classes such as the Stern numbers and so on. Among other ingredients, our proofs use results on Hankel determinants obtained recently by Han., Comment: International Mathematics Research Notices 2015

We prove that a family of average weights for oscillating tableaux are polynomials in two variables, namely, the length of the oscillating tableau and the size of the ending partition, which generalizes a result of Hopkins and Zhang. Several explicit and asymptotic formulas for the average weights are also derived., Comment: 12 pages

A multiset hook length formula for integer partitions is established by using combinatorial manipulation. As special cases, we rederive three hook length formulas, two of them obtained by Nekrasov-Okounkov, the third one by Iqbal, Nazir, Raza and Saleem, who have made use of the cyclic symmetry of the topological vertex. A multiset hook-content formula is also proved., Comment: 19 pages; 3 figures

Recently, the first author generalized a formula of Nekrasov and Okounkov which gives a combinatorial formula, in terms of hook lengths of partitions, for the coefficients of certain power series. In the course of this investigation, he conjectured that $A000731(n)=0$ if and only if $A033687(n)=0$. The numbers $A000731(n)$ are given in terms of hook lengths of partitions, while $A033687(n)$ equals the number of 3-core partitions of $n$. Here we prove this conjecture., 7 pages

The flag-major index "fmaj" and the classical length function "$\ell$" are used to construct two $q$-analogs of the generating polynomial for the hyperoctahedral group~$B_n$ by number of positive and negative fixed points (resp. pixed points). Specializations of those $q$-analogs are also derived dealing with signed derangements and desarrangements, as well as several classical results that were previously proved for the symmetric group., Comment: 21 pages

Motivated by the Nekrasov-Okounkov formula on hook lengths, the first author conjectured that the Plancherel average of the $2k$-th power sum of hook lengths of partitions with size $n$ is always a polynomial of $n$ for any $k\in \mathbb{N}$. This conjecture was generalized and proved by Stanley (Ramanujan J., 23(1--3): 91--105, 2010). In this paper, inspired by the work of Stanley and Olshanski on the differential poset of Young lattice, we study the properties of two kinds of difference operators $D$ and $D^-$ defined on functions of partitions. Even though the calculations for higher orders of $D$ are extremely complex, we prove that several well-known families of functions of partitions are annihilated by a power of the difference operator $D$. As an application, our results lead to several generalizations of classic results on partitions, including the marked hook formula, Stanley Theorem, Okada-Panova hook length formula, and Fujii-Kanno-Moriyama-Okada content formula. We insist that the Okada constants $K_r$ arise directly from the computation for a single partition $\lambda$, without the summation ranging over all partitions of size~$n$., Comment: 27 pages

International audience; The Hankel determinants of a given power series f can be evaluated by using the Jacobi continued fraction expansion of f. However the existence of the Jacobi continued fraction needs that all Hankel determinants of f are nonzero. We introduce Hankel continued fraction, whose existence and uniqueness are guaranteed without any condition for the power series f. The Hankel determinants can also be evaluated by using the Hankel continued fraction. It is well known that the continued fraction expansion of a quadratic irrational number is ultimately periodic. We prove a similar result for power series. If a power series f over a finite field satisfies a quadratic equation, then the Hankel continued fraction is ultimately periodic. As an application, we derive the Hankel determinants of several automatic sequences, in particular, the regular paperfolding sequence. Thus we provide an automatic proof of a result obtained by Guo, Wu and Wen, which was conjectured by Coons-Vrbik.

In 1998, Allouche, Peyri\`ere, Wen and Wen considered the Thue--Morse sequence, and proved that all the Hankel determinants of the period-doubling sequence are odd integral numbers. We speak of $t$-extension when the entries along the diagonal in the Hankel determinant are all multiplied by~$t$. Then we prove that the $t$-extension of each Hankel determinant of the period-doubling sequence is a polynomial in $t$, whose leading coefficient is the {\it only one} to be an odd integral number. Our proof makes use of the combinatorial set-up developed by Bugeaud and Han, which appears to be very suitable for this study, as the parameter $t$ counts the number of fixed points of a permutation. Finally, we prove that all the $t$-extensions of the Hankel determinants of the regular paperfolding sequence are polynomials in $t$ of degree less than or equal to $3$.

In 1998, Allouche, Peyrière, Wen and Wen established that the Hankel determinants associated with the Thue-Morse sequence on $\{-1,1\}$ are always nonzero. Their proof depends on a set of sixteen recurrence relations. We present an alternative, purely combinatorial proof of the same result. We also re-prove a recent result of Coons on the non-vanishing of the Hankel determinants associated to two other classical integer sequences.

A true Tree Calculus is being developed to make a joint study of the two statistics “eoc” (end of minimal chain) and “pom” (parent of maximum leaf) on the set of secant trees. Their joint distribution restricted to the set {eoc-pom ≤ 1} is shown to satisfy two partial difference equation systems, to be symmetric and to be expressed in the form of an explicit three-variable generating function.

International audience; The finite difference equation system introduced by Christiane Poupard in the study of tangent trees is reinterpreted in the alternating permutation environment. It makes it possible to make a joint study of both tangent and secant trees and calculate the generating polynomial for alternating permutations by a new statistic, referred to as being the greater neighbour of the maximum.

The classical Eulerian polynomials can be expanded in the basis $t^{k-1}(1+t)^{n+1-2k}$ ($1\leq k\leq\lfloor (n+1)/2\rfloor$) with positive integral coefficients. This formula implies both the symmetry and the unimodality of the Eulerian polynomials. In this paper, we prove a $q$-analogue of this expansion for Carlitz's $q$-Eulerian polynomials as well as a similar formula for Chow-Gessel's $q$-Eulerian polynomials of type $B$. We shall give some applications of these two formulae, which involve two new sequences of polynomials in the variable $q$ with positive integral coefficients. An open problem is to give a combinatorial interpretation for these polynomials., 13 pages, to appear in The Ramanujan Journal

The paper is devoted to the derivation of the expansion formula for the powers of the Euler Product in terms of partition hook lengths, discovered by Nekrasov and Okounkov in their study of the Seiberg-Witten Theory. We provide a refinement based on a new property of t-cores, and give an elementary proof by using the Macdonald identities. We also obtain an extension by adding two more parameters, which appears to be a discrete interpolation between the Macdonald identities and the generating function for t-cores. Several applications are derived, including the "marked hook formula"., Comment: 28 pages

Some conjectures on partition hook lengths, recently stated by the author, have been proved and generalized by Stanley, who also needed a formula by Andrews, Goulden and Jackson on symmetric functions to complete his derivation. Another identity on symmetric functions can be used instead. The purpose of this note is to prove it., Comment: 9 pages

We discover another one-parameter generalization of Postnikov's hook length formula for binary trees. The particularity of our formula is that the hook length $h_v$ appears as an exponent. As an application, we derive another simple hook length formula for binary trees when the underlying parameter takes the value 1/2., Comment: 4 pages

We construct a basis for the right quantum algebra introduced by Garoufalidis, Le and Zeilberger and give a method making it possible to go from an algebra submitted to commutation relations (without the variable q) to the right quantum algebra by means of an appropriate weight-function. As a consequence, a strong quantum MacMahon Master Theorem is derived. Besides, the algebra of biwords is systematically in use., Comment: 9 pages

Using classical transformations on the symmetric group and two transformations constructed in Fix-Mahonian Calculus I, we show that several multivariable statistics are equidistributed either with the triplet (fix,@?des,@?maj), or the pair (fix,@?maj), where ''fix,'' ''des'' and ''maj'' denote the number of fixed points, the number of descents and the major index, respectively.

We propose a new proof of the quantum version of MacMahon's Master Theorem, established by Garoufalidis, Le and Zeilberger., Comment: 8 pages

We construct two bijections of the symmetric group S"n onto itself that enable us to show that three new three-variable statistics are equidistributed with classical statistics involving the number of fixed points. The first one is equidistributed with the triplet (fix,des,maj), the last two with (fix,exc,maj), where ''fix'', ''des'', ''exc'' and ''maj'' denote the number of fixed points, the number of descents, the number of excedances and the major index, respectively.

A four-variable distribution on permutations is derived, with two dual combinatorial interpretations. The first one includes the number of fixed points "fix", the second the so-called "pix" statistic. This shows that the duality between derangements and desarrangements can be extended to the case of multivariable statistics. Several specializations are obtained, including the joint distribution of (des, exc), where "des" and "exc" stand for the number of descents and excedances, respectively., 26 pages

We extend Stanley's work on alternating permutations with extremal number of fixed points in two directions: first, alternating permutations are replaced by permutations with a prescribed descent set; second, instead of simply counting permutations we study their generating polynomials by number of excedances. Several techniques are used: Desarmenien's desarrangement combinatorics, Gessel's hook-factorization and the analytical properties of two new permutation statistics "DEZ" and "lec". Explicit formulas for the maximal case are derived by using symmetric function tools., minor change about corollary 3

We study some specializations and extensions of the quantum version of the MacMahon Master Theorem derived by Garoufalidis, Le and Zeilberger. In particular, we obtain a (t,q)-analogue for the Cartier-Foata noncommutative version and a semi-strong (t,q)-analogue for the contextual algebra., Comment: 12 pages