scholarly journals A Combinatorial Proof of the Non-Vanishing of Hankel Determinants of the Thue-Morse Sequence

10.37236/3831 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Yann Bugeaud ◽  
Guo-Niu Han
Keyword(s):  
Recent Result ◽  
Recurrence Relations ◽  
Combinatorial Proof ◽  
Integer Sequences ◽  
Hankel Determinants

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.

scholarly journals Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes

10.37236/1052 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Brad Jackson ◽  
Frank Ruskey
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Binary Trees ◽  
Integer Sequences ◽  
Number Of Leaves ◽  
Online Encyclopedia ◽  
Fibonacci Sequences ◽  
Infinite Sequences

We consider a family of meta-Fibonacci sequences which arise in studying the number of leaves at the largest level in certain infinite sequences of binary trees, restricted compositions of an integer, and binary compact codes. For this family of meta-Fibonacci sequences and two families of related sequences we derive ordinary generating functions and recurrence relations. Included in these families of sequences are several well-known sequences in the Online Encyclopedia of Integer Sequences (OEIS).

scholarly journals COUNTING PERMUTATIONS BY NUMBERS OF EXCEDANCES, FIXED POINTS AND CYCLES

2011 ◽  
Vol 85 (3) ◽  
pp. 415-421
Author(s):  
SHI-MEI MA
Keyword(s):  
Fixed Points ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Combinatorial Proof

AbstractIn this paper we present a combinatorial proof of an identity involving the two kinds of Stirling numbers and the numbers of permutations with prescribed numbers of excedances and cycles. Several recurrence relations related to the numbers of excedances, fixed points and cycles are also obtained.

scholarly journals On some results concerning the polygonal polynomials

2019 ◽  
Vol 35 (1) ◽  
pp. 01-12
Author(s):  
DORIN ANDRICA ◽  
◽  
OVIDIU BAGDASAR ◽  
Keyword(s):  
Recurrence Relations ◽  
Integer Sequences ◽  
Open Questions ◽  
New Meanings ◽  
Online Encyclopedia ◽  
Integral Formulae

In this paper we define the nth polygonal polynomial and we investigate recurrence relations and exact integral formulae for the coefficients of Pn and for those of the Mahonianpolynomials. We also explore numerical properties of these coefficients, unraveling new meanings for old sequences and generating novel entries to the Online Encyclopedia of Integer Sequences (OEIS). Some open questions are also formulated.

scholarly journals Minimal transitive factorizations of a permutation of type (p,q)

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Jang Soo Kim ◽  
Seunghyun Seo ◽  
Heesung Shin
Keyword(s):  
Recent Result ◽  
Combinatorial Proof ◽  
Type B ◽  
Noncrossing Partitions ◽  
International Audience ◽  
Maximal Chains ◽  
Nous Utilisons

International audience We give a combinatorial proof of Goulden and Jackson's formula for the number of minimal transitive factorizations of a permutation when the permutation has two cycles. We use the recent result of Goulden, Nica, and Oancea on the number of maximal chains of annular noncrossing partitions of type B. Nous donnons une preuve combinatoire de formule de Goulden et Jackson pour le nombre de factorisations transitives minimales d'une permutation lorsque la permutation a deux cycles. Nous utilisons le rèsultat rècent de Goulden, Nica, et Oancea sur le nombre de chaî nes maximales des partitions non-croisèes annulaires de type B.

scholarly journals The Generating Function of Ternary Trees and Continued Fractions

10.37236/1079 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Ira M. Gessel ◽  
Guoce Xin
Keyword(s):  
Generating Function ◽  
Continued Fraction ◽  
Hypergeometric Series ◽  
Combinatorial Proof ◽  
Hankel Determinant ◽  
Simple Method ◽  
Alternating Sign Matrices ◽  
Hankel Determinants ◽  
Fraction Method ◽  
Special Case

Michael Somos conjectured a relation between Hankel determinants whose entries ${1\over 2n+1}{3n\choose n}$ count ternary trees and the number of certain plane partitions and alternating sign matrices. Tamm evaluated these determinants by showing that the generating function for these entries has a continued fraction that is a special case of Gauss's continued fraction for a quotient of hypergeometric series. We give a systematic application of the continued fraction method to a number of similar Hankel determinants. We also describe a simple method for transforming determinants using the generating function for their entries. In this way we transform Somos's Hankel determinants to known determinants, and we obtain, up to a power of $3$, a Hankel determinant for the number of alternating sign matrices. We obtain a combinatorial proof, in terms of nonintersecting paths, of determinant identities involving the number of ternary trees and more general determinant identities involving the number of $r$-ary trees.

Integer sequences that behave as Fibonacci-Lucas pairs

2013 ◽  
Vol 97 (538) ◽  
pp. 1-7
Author(s):  
A. S. Di Domenico
Keyword(s):  
Recurrence Relations ◽  
Second Order ◽  
Integer Sequences ◽  
Hyperbolic Functions ◽  
Lucas Sequences ◽  
Constant Coefficients ◽  
Image Position

There have been a number of articles on the relation between the terms of the Fibonacci and Lucas sequences and how they are closely related to trigonometric and hyperbolic functions and their properties [1]. This article is based on other integer sequences. It sets out to determine other pairs of such sequences that have the same relation as the Fibonacci and Lucas have to each other. So we shall be concerned with second order recurrence relations with constant coefficients:and pairs of sequences (un) and (vn) that each satisfy it. We seek a condition that ensures the pair of sequences behave as the Fibonacci-Lucas pair behave.

scholarly journals Combinatorial Necklace Splitting

10.37236/168 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Dömötör Pálvölgyi
Keyword(s):  
Recent Result ◽  
Combinatorial Proof ◽  
Ulam Theorem ◽  
Splitting Problem

We give a new, combinatorial proof for the necklace splitting problem for two thieves using only Tucker's lemma (a combinatorial version of the Borsuk-Ulam theorem). We show how this method can be applied to obtain a related recent result of Simonyi and even generalize it.

scholarly journals Exact divisibility by powers of the integers in the Lucas sequences of the first and second kinds

2021 ◽  
Vol 6 (11) ◽  
pp. 11733-11748
Author(s):  
Kritkhajohn Onphaeng ◽  
◽  
Prapanpong Pongsriiam
Keyword(s):  
Recurrence Relations ◽  
Previous Article ◽  
Integer Sequences ◽  
Lucas Sequences ◽  
Positive Integers

<abstract><p>Lucas sequences of the first and second kinds are, respectively, the integer sequences $ (U_n)_{n\geq0} $ and $ (V_n)_{n\geq0} $ depending on parameters $ a, b\in\mathbb{Z} $ and defined by the recurrence relations $ U_0 = 0 $, $ U_1 = 1 $, and $ U_n = aU_{n-1}+bU_{n-2} $ for $ n\geq2 $, $ V_0 = 2 $, $ V_1 = a $, and $ V_n = aV_{n-1}+bV_{n-2} $ for $ n\geq2 $. In this article, we obtain exact divisibility results concerning $ U_n^k $ and $ V_n^k $ for all positive integers $ n $ and $ k $. This and our previous article extend many results in the literature and complete a long investigation on this problem from 1970 to 2021.</p></abstract>

scholarly journals From a Polynomial Riemann Hypothesis to Alternating Sign Matrices

10.37236/1580 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Ömer Eğecioğlu ◽  
Timothy Redmond ◽  
Charles Ryavec
Keyword(s):  
Orthogonal Polynomials ◽  
Linear Functional ◽  
Higher Order ◽  
Integer Sequences ◽  
Alternating Sign Matrices ◽  
Hankel Determinants ◽  
Real Zeros ◽  
Polynomial Sequences ◽  
Path Models ◽  
Condition C

This paper begins with a brief discussion of a class of polynomial Riemann hypotheses, which leads to the consideration of sequences of orthogonal polynomials and 3-term recursions. The discussion further leads to higher order polynomial recursions, including 4-term recursions where orthogonality is lost. Nevertheless, we show that classical results on the nature of zeros of real orthogonal polynomials (i. e., that the zeros of $p_n$ are real and those of $p_{n+1}$ interleave those of $p_n$) may be extended to polynomial sequences satisfying certain 4-term recursions. We identify specific polynomial sequences satisfying higher order recursions that should also satisfy this classical result. As with the 3-term recursions, the 4-term recursions give rise naturally to a linear functional. In the case of 3-term recursions the zeros fall nicely into place when it is known that the functional is positive, but in the case of our 4-term recursions, we show that the functional can be positive even when there are non-real zeros among some of the polynomials. It is interesting, however, that for our 4-term recursions positivity is guaranteed when a certain real parameter $C$ satisfies $C\ge 3$, and this is exactly the condition of our result that guarantees the zeros have the aforementioned interleaving property. We conjecture the condition $C\ge 3$ is also necessary. Next we used a classical determinant criterion to find exactly when the associated linear functional is positive, and we found that the Hankel determinants $\Delta_n$ formed from the sequence of moments of the functional when $C = 3$ give rise to the initial values of the integer sequence $1, 3, 26, 646, 45885, \cdots,$ of Alternating Sign Matrices (ASMs) with vertical symmetry. This spurred an intense interest in these moments, and we give 9 diverse characterizations of this sequence of moments. We then specify these Hankel determinants as Macdonald-type integrals. We also provide an an infinite class of integer sequences, each sequence of which gives the Hankel determinants $\Delta_n$ of the moments. Finally we show that certain $n$-tuples of non-intersecting lattice paths are evaluated by a related class of special Hankel determinants. This class includes the $\Delta_n$. At the same time, ASMs with vertical symmetry can readily be identified with certain $n$-tuples of osculating paths. These two lattice path models appear as a natural bridge from the ASMs with vertical symmetry to Hankel determinants.

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