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 Enumeration of Fuss-Schröder paths

10.37236/6719 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Suhyung An ◽  
JiYoon Jung ◽  
Sangwook Kim
Keyword(s):  
Combinatorial Interpretation ◽  
Noncrossing Partitions ◽  
Schröder Paths

In this paper we enumerate the number of $(k, r)$-Fuss-Schröder paths of type $\lambda$. Y. Park and S. Kim studied small Schröder paths with type $\lambda$. Generalizing the results to small $(k, r)$-Fuss-Schröder paths with type $\lambda$, we give a combinatorial interpretation for the number of small $(k, r)$-Fuss-Schröder paths of type $\lambda$ by using Chung-Feller style. We also give two sets of sparse noncrossing partitions of $[2(k + 1)n + 1]$ and $[2(k + 1)n + 2]$ which are in bijection with the set of all small and large, respectively, $(k, r)$-Fuss-Schröder paths of type $\lambda$.

scholarly journals The Generalized Schröder Theory

10.37236/1950 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Chunwei Song
Keyword(s):  
Catalan Number ◽  
Parking Functions ◽  
Combinatorial Interpretation ◽  
Higher Dimensional ◽  
Schröder Numbers ◽  
Schröder Paths ◽  
Shuffle Conjecture

While the standard Catalan and Schröder theories both have been extensively studied, people have only begun to investigate higher dimensional versions of the Catalan number (see, say, the 1991 paper of Hilton and Pedersen, and the 1996 paper of Garsia and Haiman). In this paper, we study a yet more general case, the higher dimensional Schröder theory. We define $m$-Schröder paths, find the number of such paths from $(0,0)$ to $(mn, n)$, and obtain some other results on the $m$-Schröder paths and $m$-Schröder words. Hoping to generalize classical $q$-analogue results of the ordinary Catalan and Schröder numbers, such as in the works of Fürlinger and Hofbauer, Cigler, and Bonin, Shapiro and Simion, we derive a $q$-identity which would welcome a combinatorial interpretation. Finally, we introduce the ($q, t$)-$m$-Schröder polynomial through "$m$-parking functions" and relate it to the $m$-Shuffle Conjecture of Haglund, Haiman, Loehr, Remmel and Ulyanov.

A Chung–Feller property for the generalized Schröder paths

2020 ◽  
Vol 343 (5) ◽  
pp. 111826
Author(s):  
Lin Yang ◽  
Sheng-Liang Yang
Keyword(s):  
Feller Property ◽  
Schröder Paths

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 A Combinatorial Interpretation of the Area of Schröder Paths

10.37236/1472 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
E. Pergola ◽  
R. Pinzani
Keyword(s):  
Recurrence Relation ◽  
Initial Conditions ◽  
Lattice Path ◽  
Combinatorial Interpretation ◽  
Schröder Paths

An elevated Schröder path is a lattice path that uses the steps $(1,1)$, $(1,-1)$, and $(2,0)$, that begins and ends on the $x$-axis, and that remains strictly above the $x$-axis otherwise. The total area of elevated Schröder paths of length $2n+2$ satisfies the recurrence $f_{n+1}=6f_n-f_{n-1}$, $n \geq 2$, with the initial conditions $f_0=1$, $f_1=7$. A combinatorial interpretation of this recurrence is given, by first introducing sets of unrestricted paths whose cardinality also satisfies the recurrence relation and then establishing a bijection between the set of these paths and the set of triangles constituting the total area of elevated Schröder paths.

scholarly journals Chung-Feller Property in View of Generating Functions

10.37236/591 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Shu-Chung Liu ◽  
Yi Wang ◽  
Yeong-Nan Yeh
Keyword(s):  
Functional Equation ◽  
Generating Functions ◽  
Catalan Number ◽  
Property A ◽  
New Approach ◽  
Feller Property ◽  
Uniform Partition ◽  
Schröder Paths ◽  
Partition Method

The classical Chung-Feller Theorem offers an elegant perspective for enumerating the Catalan number $c_n= \frac{1}{n+1}\binom{2n}{n}$. One of the various proofs is by the uniform-partition method. The method shows that the set of the free Dyck $n$-paths, which have $\binom{2n}{n}$ in total, is uniformly partitioned into $n+1$ blocks, and the ordinary Dyck $n$-paths form one of these blocks; therefore the cardinality of each block is $\frac{1}{n+1}\binom{2n}{n}$. In this article, we study the Chung-Feller property: a sup-structure set can be uniformly partitioned such that one of the partition blocks is (isomorphic to) a well-known structure set. The previous works about the uniform-partition method used bijections, but here we apply generating functions as a new approach. By claiming a functional equation involving the generating functions of sup- and sub-structure sets, we re-prove two known results about Chung-Feller property, and explore several new examples including the ones for the large and the little Schröder paths. Especially for the Schröder paths, we are led by the new approach straightforwardly to consider "weighted" free Schröder paths as sup-structures. The weighted structures are not obvious via bijections or other methods.

scholarly journals Generalized monotone triangles

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Lukas Riegler
Keyword(s):  
Recent Work ◽  
Computational Experiments ◽  
Combinatorial Interpretation ◽  
Alternating Sign Matrices ◽  
Round Numbers ◽  
International Audience

International audience In a recent work, the combinatorial interpretation of the polynomial $\alpha (n; k_1,k_2,\ldots,k_n)$ counting the number of Monotone Triangles with bottom row $k_1 < k_2 < ⋯< k_n$ was extended to weakly decreasing sequences $k_1 ≥k_2 ≥⋯≥k_n$. In this case the evaluation of the polynomial is equal to a signed enumeration of objects called Decreasing Monotone Triangles. In this paper we define Generalized Monotone Triangles – a joint generalization of both ordinary Monotone Triangles and Decreasing Monotone Triangles. As main result of the paper we prove that the evaluation of $\alpha (n; k_1,k_2,\ldots,k_n)$ at arbitrary $(k_1,k_2,\ldots,k_n) ∈ \mathbb{Z}^n$ is a signed enumeration of Generalized Monotone Triangles with bottom row $(k_1,k_2,\ldots,k_n)$. Computational experiments indicate that certain evaluations of the polynomial at integral sequences yield well-known round numbers related to Alternating Sign Matrices. The main result provides a combinatorial interpretation of the conjectured identities and could turn out useful in giving bijective proofs.

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 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.

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