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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

Sequences with perfect linear complexity profile were defined more than thirty years ago in the study of measures of randomness for binary sequences. More recently apwenian sequences, first with values ±1, then with values in { 0 , 1 } , were introduced in the study of Hankel determinants of automatic sequences. We explain that these two families of sequences are the same up to indexing, and give consequences and questions that this implies. We hope that this will help gathering two distinct communities of researchers.

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

This paper proposes a novel fractional transform, denoted as the fractional Krawtchouk transform (FrKT), a generalization of the Krawtchouk transform. The derivation of the FrKT uses the eigenvalue decomposition method. We determine the eigenvalues and the corresponding multiplicity of the Krawtchouk transform matrix. Moreover, the orthonormal eigenvectors of the transform matrix are derived. For validation purpose only and as a first illustration of the interest of FrKT, a watermarking example was chosen. Experimental results show that better watermark robustness and imperceptibility are achieved by adjusting the fractional orders in the FrKT.