American Institute of Mathematical Sciences

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2688-1594

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Special issue on combinatorial number Th. & enumerative combinatorics

Zhi-Wei Sun, Department of Mathematics, Nanjing University, Nanjing 210093, China zwsun@nju.edu.cn

Jiang Zeng, Institut Camille JordanUniversité Claude Bernard Lyon 143, boulevard du 11 novembre 1918F-69622 Villeurbanne Cedex, France zeng@math.univ-lyon1.fr

2020, 28(1): 559-566 doi: 10.3934/era.2020029 +[Abstract](11)+[HTML](10) +[PDF](252.97KB)
Abstract:

The pentagonal numbers are the integers given by\begin{document}$p_5(n) = n(3n-1)/2\ (n = 0,1,2,\ldots)$\end{document}.Let \begin{document}$(b,c,d)$\end{document} be one of the triples \begin{document}$(1,1,2),(1,2,3),(1,2,6)$\end{document} and \begin{document}$(2,3,4)$\end{document}.We show that each \begin{document}$n = 0,1,2,\ldots$\end{document} can be written as \begin{document}$w+bx+cy+dz$\end{document} with \begin{document}$w,x,y,z$\end{document} pentagonal numbers, which was first conjectured by Z.-W. Sun in 2016. In particular, any nonnegative integeris a sum of five pentagonal numbers two of which are equal; this refines a classical resultof Cauchy claimed by Fermat.

2020, 28(2): 1023-1030 doi: 10.3934/era.2020054 +[Abstract](23)+[HTML](12) +[PDF](324.81KB)
Abstract:

We confirm a conjectural supercongruence involving Catalan numbers, which is one of the 100 selected open conjectures on congruences of Sun. The proof makes use of hypergeometric series identities and symbolic summation method.

2020, 28(2): 1031-1036 doi: 10.3934/era.2020055 +[Abstract](22)+[HTML](11) +[PDF](258.92KB)
Abstract:

Using Watson's terminating \begin{document}$_8\phi_7$\end{document} transformation formula, we prove a family of \begin{document}$q$\end{document}-congruences modulo the square of a cyclotomic polynomial, which were originally conjectured by the author and Zudilin [J. Math. Anal. Appl. 475 (2019), 1636-646]. As an application, we deduce two supercongruences modulo \begin{document}$p^4$\end{document} (\begin{document}$p$\end{document} is an odd prime) and their \begin{document}$q$\end{document}-analogues. This also partially confirms a special case of Swisher's (H.3) conjecture.

2020, 28(3): 1273-1342 doi: 10.3934/era.2020070 +[Abstract](12)+[HTML](10) +[PDF](720.4KB)
Abstract:

Via symbolic computation we deduce 97 new type series for powers of \begin{document}$\pi$\end{document} related to Ramanujan-type series. Here are three typical examples:

with

and

where the generalized central trinomial coefficient \begin{document}$T_k(b,c)$\end{document} denotes the coefficient of \begin{document}$x^k$\end{document} in the expansion of \begin{document}$(x^2+bx+c)^k$\end{document}. We also formulate a general characterization of rational Ramanujan-type series for \begin{document}$1/\pi$\end{document} via congruences, and pose 117 new conjectural series for powers of \begin{document}$\pi$\end{document} via looking for corresponding congruences. For example, we conjecture that

Eighteen of the new series in this paper involve some imaginary quadratic fields with class number \begin{document}$8$\end{document}.

2021, 29(2): 1925-1944 doi: 10.3934/era.2020098 +[Abstract](765)+[HTML](370) +[PDF](398.24KB)
Abstract:

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.

2021, 29(2): 2167-2185 doi: 10.3934/era.2020111 +[Abstract](684)+[HTML](284) +[PDF](435.77KB)
Abstract:

The Euler number \begin{document}$E_n$\end{document} (resp. Entringer number \begin{document}$E_{n,k}$\end{document}) enumerates the alternating (down-up) permutations of \begin{document}$\{1,\dots,n\}$\end{document} (resp. starting with \begin{document}$k$\end{document}). The Springer number \begin{document}$S_n$\end{document} (resp. Arnold number \begin{document}$S_{n,k}$\end{document}) enumerates the type \begin{document}$B$\end{document} alternating permutations (resp. starting with \begin{document}$k$\end{document}). In this paper, using bijections we first derive the counterparts in André permutations and Simsun permutations for the Entringer numbers \begin{document}$(E_{n,k})$\end{document}, and then the counterparts in signed André permutations and type \begin{document}$B$\end{document} increasing 1-2 trees for the Arnold numbers \begin{document}$(S_{n,k})$\end{document}.

2021, 29(4): 2657-2671 doi: 10.3934/era.2021007 +[Abstract](406)+[HTML](229) +[PDF](346.35KB)
Abstract:

Based on Gosper's algorithm, we present an approach to the telescoping of general sequences. Along this approach, we propose a summation formula and a bibasic extension of Ma's inversion formula. From the formulas, we are able to derive several hypergeometric and elliptic hypergeometric identities.

2021, 29(5): 2877-2913 doi: 10.3934/era.2021018 +[Abstract](15)+[HTML](11) +[PDF](534.03KB)
Abstract:

2020 Impact Factor: 1.833
5 Year Impact Factor: 1.833