scholarly journals A Simple Proof of the Aztec Diamond Theorem

10.37236/1915 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Sen-Peng Eu ◽  
Tung-Shan Fu
Keyword(s):  
Simple Proof ◽  
Lattice Paths ◽  
Hankel Determinants ◽  
Domino Tilings ◽  
Aztec Diamond ◽  
Schröder Numbers

Based on a bijection between domino tilings of an Aztec diamond and non-intersecting lattice paths, a simple proof of the Aztec diamond theorem is given by means of Hankel determinants of the large and small Schröder numbers.

scholarly journals A Bijection Proving the Aztec Diamond Theorem by Combing Lattice Paths

10.37236/2809 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Frédéric Bosio ◽  
Marc A. A. Van Leeuwen
Keyword(s):  
Lattice Paths ◽  
Square Grid ◽  
Domino Tilings ◽  
Aztec Diamond ◽  
Bijective Proof ◽  
Schröder Paths ◽  
Intersecting Families ◽  
Grid Points

We give a bijective proof of the Aztec diamond theorem, stating that there are $2^{n(n+1)/2}$ domino tilings of the Aztec diamond of order $n$. The proof in fact establishes a similar result for non-intersecting families of $n+1$ Schröder paths, with horizontal, diagonal or vertical steps, linking the grid points of two adjacent sides of an $n\times n$ square grid; these families are well known to be in bijection with tilings of the Aztec diamond. Our bijection is produced by an invertible "combing'' algorithm, operating on families of paths without non-intersection condition, but instead with the requirement that any vertical steps come at the end of a path, and which are clearly $2^{n(n+1)/2}$ in number; it transforms them into non-intersecting families.

scholarly journals Hankel determinants of sums of consecutive weighted Schröder numbers

2012 ◽  
Vol 437 (9) ◽  
pp. 2285-2299 ◽  
Author(s):  
Sen-Peng Eu ◽  
Tsai-Lien Wong ◽  
Pei-Lan Yen
Keyword(s):  
Hankel Determinants ◽  
Schröder Numbers

scholarly journals Symmetric matrices, Catalan paths, and correlations

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Emmanuel Tsukerman ◽  
Lauren Williams ◽  
Bernd Sturmfels
Keyword(s):  
Laurent Polynomial ◽  
Symmetric Matrices ◽  
Correlation Matrices ◽  
Domino Tilings ◽  
Aztec Diamond ◽  
International Audience ◽  
Square Matrix

International audience Kenyon and Pemantle (2014) gave a formula for the entries of a square matrix in terms of connected principal and almost-principal minors. Each entry is an explicit Laurent polynomial whose terms are the weights of domino tilings of a half Aztec diamond. They conjectured an analogue of this parametrization for symmetric matrices, where the Laurent monomials are indexed by Catalan paths. In this paper we prove the Kenyon-Pemantle conjecture, and apply this to a statistics problem pioneered by Joe (2006). Correlation matrices are represented by an explicit bijection from the cube to the elliptope.

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 Domino tilings of the expanded Aztec diamond

2018 ◽  
Vol 341 (4) ◽  
pp. 1185-1191 ◽  
Author(s):  
Seungsang Oh
Keyword(s):  
Domino Tilings ◽  
Aztec Diamond

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.

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 Simple Proof for the Number of Tilings of Quartered Aztec Diamonds

10.37236/3429 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Tri Lai
Keyword(s):  
Simple Proof ◽  
Aztec Diamond ◽  
Unpublished Work ◽  
Simple Product ◽  
Product Formulas

We get four quartered Aztec diamonds by dividing an Aztec diamond region by two zigzag cuts passing its center. W. Jockusch and J. Propp (in an unpublished work) found that the number of tilings of quartered Aztec diamonds is given by simple product formulas. In this paper we present a simple proof for this result.

scholarly journals Hankel determinants of linear combinations of moments of orthogonal polynomials, II

2021 ◽  
Author(s):  
C. Krattenthaler
Keyword(s):  
Orthogonal Polynomials ◽  
Matrix Theory ◽  
Binomial Coefficients ◽  
Linear Recurrence ◽  
Hankel Determinants ◽  
Motzkin Numbers ◽  
Schröder Numbers ◽  
Delannoy Numbers ◽  
Dodgson Condensation ◽  
Full Proof

AbstractWe present a formula that expresses the Hankel determinants of a linear combination of length $$d+1$$ d + 1 of moments of orthogonal polynomials in terms of a $$d\times d$$ d × d determinant of the orthogonal polynomials. This formula exists somehow hidden in the folklore of the theory of orthogonal polynomials but deserves to be better known, and be presented correctly and with full proof. We present four fundamentally different proofs, one that uses classical formulae from the theory of orthogonal polynomials, one that uses a vanishing argument and is due to Elouafi (J Math Anal Appl 431:1253–1274, 2015) (but given in an incomplete form there), one that is inspired by random matrix theory and is due to Brézin and Hikami (Commun Math Phys 214:111–135, 2000), and one that uses (Dodgson) condensation. We give two applications of the formula. In the first application, we explain how to compute such Hankel determinants in a singular case. The second application concerns the linear recurrence of such Hankel determinants for a certain class of moments that covers numerous classical combinatorial sequences, including Catalan numbers, Motzkin numbers, central binomial coefficients, central trinomial coefficients, central Delannoy numbers, Schröder numbers, Riordan numbers, and Fine numbers.

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