A Method for Determining the Mod-$2^k$ Behaviour of Recursive Sequences, with Applications to Subgroup Counting

We present a method to obtain congruences modulo powers of $2$ for sequences given by recurrences of finite depth with polynomial coefficients. We apply this method to Catalan numbers, Fuß--Catalan numbers, and to subgroup counting functions associated with Hecke groups and their lifts. This leads to numerous new results, including many extensions of known results to higher powers of $2$.

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Associative spectra of graph algebras I

Journal of Algebraic Combinatorics ◽

10.1007/s10801-020-01010-w ◽

2021 ◽

Author(s):

Erkko Lehtonen ◽

Tamás Waldhauser

Keyword(s):

Catalan Numbers ◽

Graph Algebras ◽

Undirected Graphs ◽

Powers Of 2

AbstractAssociative spectra of graph algebras are examined with the help of homomorphisms of DFS trees. Undirected graphs are classified according to the associative spectra of their graph algebras; there are only three distinct possibilities: constant 1, powers of 2, and Catalan numbers. Associative and antiassociative digraphs are described, and associative spectra are determined for certain families of digraphs, such as paths, cycles, and graphs on two vertices.

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D-finite numbers

International Journal of Number Theory ◽

10.1142/s1793042118501099 ◽

2018 ◽

Vol 14 (07) ◽

pp. 1827-1848

Author(s):

Hui Huang ◽

Manuel Kauers

Keyword(s):

Complex Numbers ◽

Algebraic Numbers ◽

Recurrence Equations ◽

Polynomial Coefficients ◽

Recursive Sequences ◽

Linear Differential ◽

Definition Of ◽

Mathematical Constants

D-finite functions and P-recursive sequences are defined in terms of linear differential and recurrence equations with polynomial coefficients. In this paper, we introduce a class of numbers closely related to D-finite functions and P-recursive sequences. It consists of the limits of convergent P-recursive sequences. Typically, this class contains many well-known mathematical constants in addition to the algebraic numbers. Our definition of the class of D-finite numbers depends on two subrings of the field of complex numbers. We investigate how different choices of these two subrings affect the class. Moreover, we show that D-finite numbers are essentially limits of D-finite functions at the point one, and evaluating D-finite functions at non-singular algebraic points typically yields D-finite numbers. This result makes it easier to recognize certain numbers to be D-finite.

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On the Catalan Numbers and Some of Their Identities

Symmetry ◽

10.3390/sym11010062 ◽

2019 ◽

Vol 11 (1) ◽

pp. 62 ◽

Cited By ~ 1

Author(s):

Wenpeng Zhang ◽

Li Chen

Keyword(s):

Catalan Numbers ◽

Recursive Sequences ◽

Combinatorial Methods

The main purpose of this paper is using the elementary and combinatorial methods to study the properties of the Catalan numbers, and give two new identities for them. In order to do this, we first introduce two new recursive sequences, then with the help of these sequences, we obtained the identities for the convolution involving the Catalan numbers.

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On the Parity of Certain Coefficients for a $q$-Analogue of the Catalan Numbers

The Electronic Journal of Combinatorics ◽

10.37236/2313 ◽

2012 ◽

Vol 19 (4) ◽

Cited By ~ 1

Author(s):

Kendra Killpatrick

Keyword(s):

Equivalence Classes ◽

Catalan Numbers ◽

Major Index ◽

Powers Of 2 ◽

Set Of Permutations ◽

Adic Valuation ◽

Permutation Statistic ◽

Wilf Equivalence

The 2-adic valuation (highest power of 2) dividing the well-known Catalan numbers, $C_n$, has been completely determined by Alter and Kubota and further studied combinatorially by Deutsch and Sagan. In particular, it is well known that $C_n$ is odd if and only if $n = 2^k-1$ for some $k \geq 0$. The polynomial $F_n^{ch}(321;q) = \sum_{\sigma \in Av_n(321)} q^{ch(\sigma)}$, where $Av_n(321)$ is the set of permutations in $S_n$ that avoid 321 and $ch$ is the charge statistic, is a $q$-analogue of the Catalan numbers since specializing $q=1$ gives $C_n$. We prove that the coefficient of $q^i$ in $F_{2^k-1}^{ch}(321;q)$ is even if $i \geq 1$, giving a refinement of the "if" direction of the $C_n$ parity result. Furthermore, we use a bijection between the charge statistic and the major index to prove a conjecture of Dokos, Dwyer, Johnson, Sagan and Selsor regarding powers of 2 and the major index. In addition, Sagan and Savage have recently defined a notion of $st$-Wilf equivalence for any permutation statistic $st$ and any two sets of permutations $\Pi$ and $\Pi'$. We say $\Pi$ and $\Pi'$ are $st$-Wilf equivalent if $\sum_{\sigma \in Av_n(\Pi)} q^{st(\sigma)} = \sum_{\sigma \in Av_n(\Pi')} q^{st(\sigma)}$. In this paper we show how one can characterize the charge-Wilf equivalence classes for subsets of $S_3$.

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