scholarly journals Counting Segmented Permutations Using Bicoloured Dyck Paths

10.37236/1936 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Anders Claesson
Keyword(s):  
Generating Function ◽  
Chebyshev Polynomials ◽  
Dyck Path ◽  
Dyck Paths ◽  
One To One ◽  
Bivariate Generating Function

A bicoloured Dyck path is a Dyck path in which each up-step is assigned one of two colours, say, red and green. We say that a permutation $\pi$ is $\sigma$-segmented if every occurrence $o$ of $\sigma$ in $\pi$ is a segment-occurrence (i.e., $o$ is a contiguous subword in $\pi$). We show combinatorially the following two results: The $132$-segmented permutations of length $n$ with $k$ occurrences of $132$ are in one-to-one correspondence with bicoloured Dyck paths of length $2n-4k$ with $k$ red up-steps. Similarly, the $123$-segmented permutations of length $n$ with $k$ occurrences of $123$ are in one-to-one correspondence with bicoloured Dyck paths of length $2n-4k$ with $k$ red up-steps, each of height less than $2$. We enumerate the permutations above by enumerating the corresponding bicoloured Dyck paths. More generally, we present a bivariate generating function for the number of bicoloured Dyck paths of length $2n$ with $k$ red up-steps, each of height less than $h$. This generating function is expressed in terms of Chebyshev polynomials of the second kind.

scholarly journals Short note on the number of 1-ascents in dispersed Dyck paths

2017 ◽  
Vol 09 (06) ◽  
pp. 1750077
Author(s):  
Kairi Kangro ◽  
Mozhgan Pourmoradnasseri ◽  
Dirk Oliver Theis
Keyword(s):  
Generating Function ◽  
Short Note ◽  
Lattice Path ◽  
Dyck Path ◽  
Closed Formula ◽  
Dyck Paths ◽  
Maximal Sequence

A dispersed Dyck path (DDP) of length [Formula: see text] is a lattice path on [Formula: see text] from [Formula: see text] to [Formula: see text] in which the following steps are allowed: “up” [Formula: see text]; “down” [Formula: see text]; and “right” [Formula: see text]. An ascent in a DDP is an inclusion-wise maximal sequence of consecutive up steps. A 1-ascent is an ascent consisting of exactly 1 up step. We give a closed formula for the total number of 1-ascents in all dispersed Dyck paths of length [Formula: see text], #A191386 in Sloane’s OEIS. Previously, only implicit generating function relations and asymptotics were known.

scholarly journals General Results on the Enumeration of Strings in Dyck Paths

10.37236/561 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
K. Manes ◽  
A. Sapounakis ◽  
I. Tasoulas ◽  
P. Tsikouras
Keyword(s):  
Generating Function ◽  
Lattice Path ◽  
Dyck Path ◽  
Dyck Paths

Let $\tau$ be a fixed lattice path (called in this context string) on the integer plane, consisting of two kinds of steps. The Dyck path statistic "number of occurrences of $\tau$" has been studied by many authors, for particular strings only. In this paper, arbitrary strings are considered. The associated generating function is evaluated when $\tau$ is a Dyck prefix (or a Dyck suffix). Furthermore, the case when $\tau$ is neither a Dyck prefix nor a Dyck suffix is considered, giving some partial results. Finally, the statistic "number of occurrences of $\tau$ at height at least $j$" is considered, evaluating the corresponding generating function when $\tau$ is either a Dyck prefix or a Dyck suffix.

scholarly journals Bijections between Directed Animals, Multisets and Grand-Dyck Paths

10.37236/8826 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Jean-Luc Baril ◽  
David Bevan ◽  
Sergey Kirgizov
Keyword(s):  
Generating Function ◽  
Square Lattice ◽  
Dyck Paths ◽  
Bivariate Generating Function

An $n$-multiset of $[k]=\{1,2,\ldots, k\}$ consists of a set of $n$ elements from $[k]$ where each element can be repeated. We present the bivariate generating function for $n$-multisets of $[k]$ with no consecutive elements. For $n=k$, these multisets have the same enumeration as directed animals in the square lattice. Then we give constructive bijections between directed animals, multisets with no consecutive elements and Grand-Dyck paths avoiding the pattern $DUD$, and we show how  classical and novel statistics are transported by these bijections.

scholarly journals On the number of restricted Dyck paths

Filomat ◽  
2011 ◽  
Vol 25 (3) ◽  
pp. 191-201
Author(s):  
Aleksandar Ilic ◽  
Andreja Ilic
Keyword(s):  
Lower Bound ◽  
Generating Function ◽  
Chebyshev Polynomials ◽  
Integer Lattice ◽  
Lattice Paths ◽  
Mathematical Society ◽  
Spectral Moments ◽  
Estrada Index ◽  
Dyck Paths ◽  
Combinatorial Technique

In this note we examine the number of integer lattice paths consisting of up-steps (1, 1) and down-steps (1,?1) that do not touch the lines y = m and y=?k, and in particular Theorem 3.2 in [P. Mladenovic, Combinatorics, Mathematical Society of Serbia, Belgrade, 2001]. The theorem is shown to be incorrect for n ? m + k + min(m,k), and using similar combinatorial technique we proved the upper and lower bound for the number of such restricted Dyck paths. In conclusion, we present some relations between the Chebyshev polynomials of the second kind and generating function for the number of restricted Dyck paths, and connections with the spectral moments of graphs and the Estrada index.

scholarly journals Bicoloured Dyck Paths and the Contact Polynomial for $n$ Non-Intersecting Paths in a Half-Plane Lattice

10.37236/1728 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
R. Brak ◽  
J. W. Essam
Keyword(s):  
Recurrence Relation ◽  
Half Plane ◽  
Generating Function ◽  
Catalan Numbers ◽  
Catalan Number ◽  
Lattice Paths ◽  
Product Form ◽  
Dyck Path ◽  
Combinatorial Interpretation ◽  
Dyck Paths

In this paper configurations of $n$ non-intersecting lattice paths which begin and end on the line $y=0$ and are excluded from the region below this line are considered. Such configurations are called Hankel $n-$paths and their contact polynomial is defined by $\hat{Z}^{\cal{H}}_{2r}(n;\kappa)\equiv \sum_{c= 1}^{r+1} |{\cal H}_{2r}^{(n)}(c)|\kappa^c$ where ${\cal H}_{2r}^{(n)}(c)$ is the set of Hankel $n$-paths which make $c$ intersections with the line $y=0$ the lowest of which has length $2r$. These configurations may also be described as parallel Dyck paths. It is found that replacing $\kappa$ by the length generating function for Dyck paths, $\kappa(\omega) \equiv \sum_{r=0}^\infty C_r \omega^r$, where $C_r$ is the $r^{th}$ Catalan number, results in a remarkable simplification of the coefficients of the contact polynomial. In particular it is shown that the polynomial for configurations of a single Dyck path has the expansion $\hat{Z}^{\cal{H}}_{2r}(1;\kappa(\omega)) = \sum_{b=0}^\infty C_{r+b}\omega^b$. This result is derived using a bijection between bi-coloured Dyck paths and plain Dyck paths. A bi-coloured Dyck path is a Dyck path in which each edge is coloured either red or blue with the constraint that the colour can only change at a contact with the line $y=0$. For $n>1$, the coefficient of $\omega^b$ in $\hat{Z}^{\cal{W}}_{2r}(n;\kappa(\omega))$ is expressed as a determinant of Catalan numbers which has a combinatorial interpretation in terms of a modified class of $n$ non-intersecting Dyck paths. The determinant satisfies a recurrence relation which leads to the proof of a product form for the coefficients in the $\omega$ expansion of the contact polynomial.

PARITY RESULTS FOR PARTITIONS WHEREIN EACH PART APPEARS AN ODD NUMBER OF TIMES

2018 ◽  
Vol 99 (1) ◽  
pp. 51-55
Author(s):  
MICHAEL D. HIRSCHHORN ◽  
JAMES A. SELLERS
Keyword(s):  
Generating Function ◽  
Proof Techniques ◽  
One To One

We consider the function $f(n)$ that enumerates partitions of weight $n$ wherein each part appears an odd number of times. Chern [‘Unlimited parity alternating partitions’, Quaest. Math. (to appear)] noted that such partitions can be placed in one-to-one correspondence with the partitions of $n$ which he calls unlimited parity alternating partitions with smallest part odd. Our goal is to study the parity of $f(n)$ in detail. In particular, we prove a characterisation of $f(2n)$ modulo 2 which implies that there are infinitely many Ramanujan-like congruences modulo 2 satisfied by the function $f.$ The proof techniques are elementary and involve classical generating function dissection tools.

scholarly journals A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path

2012 ◽  
Vol 64 (4) ◽  
pp. 822-844 ◽  
Author(s):  
J. Haglund ◽  
J. Morse ◽  
M. Zabrocki
Keyword(s):  
Building Blocks ◽  
Main Diagonal ◽  
Combinatorial Theory ◽  
Dyck Path ◽  
Dyck Paths ◽  
Littlewood Polynomials ◽  
Diagonal Harmonics ◽  
Littlewood Polynomial ◽  
Theory Operator ◽  
Shuffle Conjecture

Abstract We introduce a q, t-enumeration of Dyck paths that are forced to touch the main diagonal at specific points and forbidden to touch elsewhere and conjecture that it describes the action of the Macdonald theory ∇ operator applied to a Hall–Littlewood polynomial. Our conjecture refines several earlier conjectures concerning the space of diagonal harmonics including the “shuffle conjecture” (Duke J. Math. 126 (2005), pp. 195 − 232) for ∇ en[X]. We bring to light that certain generalized Hall–Littlewood polynomials indexed by compositions are the building blocks for the algebraic combinatorial theory of q, t-Catalan sequences, and we prove a number of identities involving these functions.

scholarly journals On a Class of Distributions Stable Under Random Summation

2012 ◽  
Vol 49 (02) ◽  
pp. 303-318 ◽  
Author(s):  
L. B. Klebanov ◽  
A. V. Kakosyan ◽  
S. T. Rachev ◽  
G. Temnov
Keyword(s):  
Normal Distribution ◽  
Generating Function ◽  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Analytic Functions ◽  
Random Variable ◽  
Random Summation ◽  
The Stability ◽  
Rational Generating Functions ◽  
Family Of Distributions

We study a family of distributions that satisfy the stability-under-addition property, provided that the number ν of random variables in a sum is also a random variable. We call the corresponding property ν-stability and investigate the situation when the semigroup generated by the generating function of ν is commutative. Using results from the theory of iterations of analytic functions, we describe ν-stable distributions generated by summations with rational generating functions. A new case in this class of distributions arises when generating functions are linked with Chebyshev polynomials. The analogue of normal distribution corresponds to the hyperbolic secant distribution.

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