scholarly journals Bijective Recurrences concerning Schröder Paths

10.37236/1385 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Robert A. Sulanke
Keyword(s):  
Lattice Paths ◽  
Schröder Numbers ◽  
Schröder Paths

Consider lattice paths in Z$^2$ with three step types: the up diagonal $(1,1)$, the down diagonal $(1,-1)$, and the double horizontal $(2,0)$. For $n \geq 1$, let $S_n$ denote the set of such paths running from $(0,0)$ to $(2n,0)$ and remaining strictly above the x-axis except initially and terminally. It is well known that the cardinalities, $r_n = |S_n|$, are the large Schröder numbers. We use lattice paths to interpret bijectively the recurrence $ (n+1) r_{n+1} = 3(2n - 1) r_{n} - (n-2) r_{n-1}$, for $n \geq 2$, with $r_1=1$ and $r_2=2$. We then use the bijective scheme to prove a result of Kreweras that the sum of the areas of the regions lying under the paths of $S_n$ and above the x-axis, denoted by $AS_n$, satisfies $ AS_{n+1} = 6 AS_n - AS_{n-1}, $ for $n \geq 2$, with $AS_1 =1$, and $AS_2 =7$. Hence $AS_n = 1, 7, 41, 239 ,1393, \ldots$. The bijective scheme yields analogous recurrences for elevated Catalan paths.

scholarly journals A Bijection on Classes Enumerated by the Schröder Numbers

2017 ◽  
Vol Vol. 18 no. 2, Permutation... (Permutation Patterns) ◽  
Author(s):  
Michael W. Schroeder ◽  
Rebecca Smith
Keyword(s):  
Lattice Paths ◽  
Sorting Machine ◽  
Schröder Numbers ◽  
Schröder Paths ◽  
In Series

We consider a sorting machine consisting of two stacks in series where the first stack has the added restriction that entries in the stack must be in decreasing order from top to bottom. The class of permutations sortable by this machine are known to be enumerated by the Schröder numbers. In this paper, we give a bijection between these sortable permutations of length $n$ and Schröder paths -- the lattice paths from $(0,0)$ to $(n-1,n-1)$ composed of East steps $(1,0)$, North steps $(0,1)$, and Diagonal steps $(1,1)$ that travel weakly below the line $y=x$.

scholarly journals Chung-Feller Property of Schröder Objects

10.37236/5659 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Youngja Park ◽  
Sangwook Kim
Keyword(s):  
Connected Components ◽  
Combinatorial Interpretation ◽  
Noncrossing Partitions ◽  
Alternating Sign Matrices ◽  
Feller Property ◽  
Bijective Proof ◽  
Schröder Numbers ◽  
Schröder Paths

Large Schröder paths, sparse noncrossing partitions, partial horizontal strips, and $132$-avoiding alternating sign matrices are objects enumerated by Schröder numbers. In this paper we give formula for the number of Schröder objects with given type and number of connected components. The proofs are bijective using Chung-Feller style. A bijective proof for the number of Schröder objects with given type is provided. We also give a combinatorial interpretation for the number of small Schröder paths.

scholarly journals Slicings of Parallelogram Polyominoes: Catalan, Schröder, Baxter, and Other Sequences

10.37236/7375 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Nicholas R. Beaton ◽  
Mathilde Bouvel ◽  
Veronica Guerrini ◽  
Simone Rinaldi
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Kernel Method ◽  
Lattice Paths ◽  
Schröder Numbers ◽  
Special Cases ◽  
Generating Tree

We provide a new succession rule (i.e. generating tree) associated with Schröder numbers, that interpolates between the known succession rules for Catalan and Baxter numbers. We define Schröder and Baxter generalizations of parallelogram polyominoes, called slicings, which grow according to these succession rules. In passing, we also exhibit Schröder subclasses of Baxter classes, namely a Schröder subset of triples of non-intersecting lattice paths, a new Schröder subset of Baxter permutations, and a new Schröder subset of mosaic floorplans. Finally, we define two families of subclasses of Baxter slicings: the $m$-skinny slicings and the $m$-row-restricted slicings, for $m \in \mathbb{N}$. Using functional equations and the kernel method, their generating functions are computed in some special cases, and we conjecture that they are algebraic for any $m$.

scholarly journals Matchings Avoiding Partial Patterns

10.37236/1138 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
William Y. C. Chen ◽  
Toufik Mansour ◽  
Sherry H. F. Yan
Keyword(s):  
Catalan Numbers ◽  
Schröder Numbers ◽  
Schröder Paths ◽  
Generating Trees ◽  
Super Catalan Numbers ◽  
Wilf Equivalence

We show that matchings avoiding a certain partial pattern are counted by the $3$-Catalan numbers. We give a characterization of $12312$-avoiding matchings in terms of restrictions on the corresponding oscillating tableaux. We also find a bijection between matchings avoiding both patterns $12312$ and $121323$ and Schröder paths without peaks at level one, which are counted by the super-Catalan numbers or the little Schröder numbers. A refinement of the super-Catalan numbers is derived by fixing the number of crossings in the matchings. In the sense of Wilf-equivalence, we use the method of generating trees to show that the patterns 12132, 12123, 12321, 12231, 12213 are all equivalent to the pattern $12312$.

scholarly journals Vertically Constrained Motzkin-Like Paths Inspired by Bobbin Lace

10.37236/7799 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Veronika Irvine ◽  
Stephen Melczer ◽  
Frank Ruskey
Keyword(s):  
Mathematical Model ◽  
Generating Functions ◽  
Finite Lattice ◽  
Lattice Paths ◽  
Change Direction ◽  
Quarter Plane ◽  
Directed Paths ◽  
Schröder Paths ◽  
Motzkin Paths

Inspired by a new mathematical model for bobbin lace, this paper considers finite lattice paths formed from the set of step vectors $\mathfrak{A}=$$\{\rightarrow,$ $\nearrow,$ $\searrow,$ $\uparrow,$ $\downarrow\}$ with the restriction that vertical steps $(\uparrow, \downarrow)$ cannot be consecutive. The set $\mathfrak{A}$ is the union of the well known Motzkin step vectors $\mathfrak{M}=$$\{\rightarrow,$ $\nearrow,$ $\searrow\}$ with the vertical steps $\{\uparrow, \downarrow\}$. An explicit bijection $\phi$ between the exhaustive set of vertically constrained paths formed from $\mathfrak{A}$ and a bisection of the paths generated by $\mathfrak{M}S$ is presented. In a similar manner, paths with the step vectors $\mathfrak{B}=$$\{\nearrow,$ $\searrow,$ $\uparrow,$ $\downarrow\}$, the union of Dyck step vectors and constrained vertical steps, are examined.  We show, using the same $\phi$ mapping, that there is a bijection between vertically constrained $\mathfrak{B}$ paths and the subset of Motzkin paths avoiding horizontal steps at even indices.  Generating functions are derived to enumerate these vertically constrained, partially directed paths when restricted to the half and quarter-plane.  Finally, we extend Schröder and Delannoy step sets in a similar manner and find a bijection between these paths and a subset of Schröder paths that are smooth (do not change direction) at a regular horizontal interval.

scholarly journals Generalizing Narayana and Schröder Numbers to Higher Dimensions

10.37236/1807 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Robert A. Sulanke
Keyword(s):  
Higher Dimensions ◽  
Lattice Paths ◽  
Dimensional Lattice ◽  
Schröder Numbers

Let ${\cal C}(d,n)$ denote the set of $d$-dimensional lattice paths using the steps $X_1 := (1, 0, \ldots, 0),$ $ X_2 := (0, 1, \ldots, 0),$ $\ldots,$ $ X_d := (0,0, \ldots,1)$, running from $(0,\ldots,0)$ to $(n,\ldots,n)$, and lying in $\{(x_1,x_2, \ldots, x_d) : 0 \le x_1 \le x_2 \le \ldots \le x_d \}$. On any path $P:=p_1p_2 \ldots p_{dn} \in {\cal C}(d,n)$, define the statistics ${\rm asc}(P) := $$|\{i : p_ip_{i+1} = X_jX_{\ell}, j < \ell \}|$ and ${\rm des}(P) := $$|\{i : p_ip_{i+1} = X_jX_{\ell}, j>\ell \}|$. Define the generalized Narayana number $N(d,n,k)$ to count the paths in ${\cal C}(d,n)$ with ${\rm asc}(P)=k$. We consider the derivation of a formula for $N(d,n,k)$, implicit in MacMahon's work. We examine other statistics for $N(d,n,k)$ and show that the statistics ${\rm asc}$ and ${\rm des}-d+1$ are equidistributed. We use Wegschaider's algorithm, extending Sister Celine's (Wilf-Zeilberger) method to multiple summation, to obtain recurrences for $N(3,n,k)$. We introduce the generalized large Schröder numbers $(2^{d-1}\sum_k N(d,n,k)2^k)_{n\ge1}$ to count constrained paths using step sets which include diagonal steps.

The m-Schröder paths and m-Schröder numbers

2021 ◽  
Vol 344 (2) ◽  
pp. 112209
Author(s):  
Sheng-Liang Yang ◽  
Mei-yang Jiang
Keyword(s):  
Schröder Numbers ◽  
Schröder Paths

scholarly journals Generalized Small Schröder Numbers

10.37236/4827 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
JiSun Huh ◽  
SeungKyung Park
Keyword(s):  
Lattice Path ◽  
Dyck Paths ◽  
Schröder Numbers ◽  
Schröder Paths ◽  
Positive Integers

We study generalized small Schröder paths in the sense of arbitrary sizes of steps. A generalized small Schröder path is a generalized lattice path from $(0,0)$ to $(2n,0)$ with the step set of  $\{(k,k), (l,-l), (2r,0)\, |\, k,l,r \in {\bf P}\}$, where ${\bf P}$ is the set of positive integers, which never goes below the $x$-axis, and with no horizontal steps at level 0.  We find a bijection between 5-colored Dyck paths and generalized small Schröder paths, proving that the number of generalized small Schröder paths is equal to $\sum_{k=1}^{n} N(n,k)5^{n-k}$ for $n\geq 1$.

scholarly journals A Schröder Generalization of Haglund's Statistic on Catalan Paths

10.37236/1709 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
E. S. Egge ◽  
J. Haglund ◽  
K. Killpatrick ◽  
D. Kremer
Keyword(s):  
Symmetric Functions ◽  
Rational Functions ◽  
Lattice Paths ◽  
Nabla Operator ◽  
Special Values ◽  
Schröder Paths ◽  
Macdonald Symmetric Functions

Garsia and Haiman (J. Algebraic. Combin. $\bf5$ $(1996)$, $191-244$) conjectured that a certain sum $C_n(q,t)$ of rational functions in $q,t$ reduces to a polynomial in $q,t$ with nonnegative integral coefficients. Haglund later discovered (Adv. Math., in press), and with Garsia proved (Proc. Nat. Acad. Sci. ${\bf98}$ $(2001)$, $4313-4316$) the refined conjecture $C_n(q,t) = \sum q^{{\rm area}}t^{{\rm bounce}}$. Here the sum is over all Catalan lattice paths and ${\rm area}$ and ${\rm bounce}$ have simple descriptions in terms of the path. In this article we give an extension of $({\rm area},{\rm bounce})$ to Schröder lattice paths, and introduce polynomials defined by summing $q^{{\rm area}}t^{{\rm bounce}}$ over certain sets of Schröder paths. We derive recurrences and special values for these polynomials, and conjecture they are symmetric in $q,t$. We also describe a much stronger conjecture involving rational functions in $q,t$ and the $\nabla$ operator from the theory of Macdonald symmetric functions.

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