Let $r(n,k)$ (resp. $s(n,k)$) be the number of Schröder paths (resp. little Schröder paths) of length $2n$ with $k$ hills, and set $r(0,0)=s(0,0)=1$. We bijectively establish the following recurrence relations: \begin{align*} r(n,0)&=\sum\limits_{j=0}^{n-1}2^{j}r(n-1,j), r(n,k)&=r(n-1,k-1)+\sum\limits_{j=k}^{n-1}2^{j-k}r(n-1,j),\quad 1\le k\le n, s(n,0) &=\sum\limits_{j=1}^{n-1}2\cdot3^{j-1}s(n-1,j), s(n,k) &=s(n-1,k-1)+\sum\limits_{j=k+1}^{n-1}2\cdot3^{j-k-1}s(n-1,j),\quad 1\le k\le n. \end{align*} The infinite lower triangular matrices $[r(n,k)]_{n,k\ge 0}$ and $[s(n,k)]_{n,k\ge 0}$, whose row sums produce the large and little Schröder numbers respectively, are two Riordan arrays of Bell type. Hence the above recurrences can also be deduced from their $A$- and $Z$-sequences characterizations.
On the other hand, it is well-known that the large Schröder numbers also enumerate separable permutations. This propelled us to reveal the connection with a lesser-known permutation statistic, called initial ascending run, whose distribution on separable permutations is shown to be given by $[r(n,k)]_{n,k\ge 0}$ as well.
It is well known that the derangement numbers $d_n$, which count permutations of length $n$ with no fixed points, satisfy the recurrence $d_n=nd_{n-1}+(-1)^n$ for $n\ge1$. Combinatorial proofs of this formula have been given by Remmel, Wilf, Désarménien and Benjamin--Ornstein. Here we present yet another, arguably simpler, bijective proof.
We provide a bijective proof of a formula of Auli and the author expressing the number of inversion sequences with no three consecutive equal entries in terms of the number of non-derangements, that is, permutations with fixed points. Additionally, we give bijective proofs of two simple recurrences for the number of non-derangements.
The absolute value of the coefficient of $q$ in the chromatic polynomial of a graph $G$ is known as the chromatic discriminant of $G$ and is denoted $α(G)$. There is a well known recurrence formula for $α(G)$ that comes from the deletion-contraction rule for the chromatic polynomial. In this paper we prove another recurrence formula for $α(G)$ that comes from the theory of Kac-Moody Lie algebras. We start with a brief survey on many interesting algebraic and combinatorial interpretations of $α(G)$. We use two of these interpretations (in terms of acyclic orientations and spanning trees) to giv…
Correlation functions in a dynamic quartic matrix model are obtained from the two-point function through a recurrence relation. This paper gives the explicit solution of the recurrence by mapping it bijectively to a two-fold nested combinatorial structure each counted by Catalan numbers. These `nested Catalan tables' have a description as diagrams of non-crossing chords and threads.
We prove that any distributional limit of finite planar graphs in which the degree of the root has an exponential tail is almost surely recurrent. As a corollary, we obtain that the uniform infinite planar triangulation and quadrangulation (UIPT and UIPQ) are almost surely recurrent, resolving a conjecture of Angel, Benjamini and Schramm. We also settle another related problem of Benjamini and Schramm. We show that in any bounded degree, finite planar graph the probability that the simple random walk started at a uniform random vertex avoids its initial location for T steps is at most C/log T.
We establish a simple recurrence formula for the number $Q_g^n$ of rooted orientable maps counted by edges and genus. We also give a weighted variant for the generating polynomial $Q_g^n(x)$ where $x$ is a parameter taking the number of faces of the map into account, or equivalently a simple recurrence formula for the refined numbers $M_g^{i,j}$ that count maps by genus, vertices, and faces. These formulas give by far the fastest known way of computing these numbers, or the fixed-genus generating functions, especially for large $g$. In the very particular case of one-face maps, we recover the …
We give a new interpretation of the derangement numbers d_n as the sum of the values of the largest fixed points of all non-derangements of length n-1. We also show that the analogous sum for the smallest fixed points equals the number of permutations of length n with at least two fixed points. We provide analytic and bijective proofs of both results, as well as a new recurrence for the derangement numbers.
A recurrence relation of the generating function of the dimer model of Fibonacci type gives a functional relation for formal power series associated to lattice paths such as a Dyck, Motzkin and Schröder path. In this paper, we generalize the correspondence to the case of generalized lattice paths, $k$-Dyck, $k$-Motzkin and $k$-Schröder paths, by modifying the recurrence relation of the dimer model. We introduce five types of generalizations of the dimer model by keeping its combinatorial structures. This allows us to express the generating functions in terms of generalized lattice paths. The w…
We study $ν$-Schröder paths, which are Schröder paths which stay weakly above a given lattice path $ν$. Some classical bijective and enumerative results are extended to the $ν$-setting, including the relationship between small and large Schröder paths. We introduce two posets of $ν$-Schröder objects, namely $ν$-Schröder paths and trees, and show that they are isomorphic to the face poset of the $ν$-associahedron $A_ν$ introduced by Ceballos, Padrol and Sarmiento. A consequence of our results is that the $i$-dimensional faces of $A_ν$ are indexed by $ν$-Schröder paths with $i$ diagonal steps, a…
Riordan paths are Motzkin paths without horizontal steps on the x-axis. We establish a correspondence between Riordan paths and $(321,3\bar{1}42)$-avoiding derangements. We also present a combinatorial proof of a recurrence relation for the Riordan numbers in the spirit of the Foata-Zeilberger proof of a recurrence relation on the Schröder numbers.
We set the foundation for a series of works aimed at proving strong relations between uniform random planar maps and Liouville quantum gravity (LQG). Our method relies on a bijective encoding of site-percolated planar triangulations by certain 2D lattice paths. Our bijection parallels in the discrete setting the \emph{mating-of-trees} framework of LQG and Schramm-Loewner evolutions (SLE) introduced by Duplantier, Miller, and Sheffield. Combining these two correspondences allows us to relate uniform site-percolated triangulations to $\sqrt{8/3}$-LQG and SLE$_6$. In particular, we establish the …
In this paper we study the number of humps (peaks) in Dyck, Motzkin and Schröder paths. Recently A. Regev noticed that the number of peaks in all Dyck paths of order $n$ is one half of the number of super Dyck paths of order $n$. He also computed the number of humps in Motzkin paths and found a similar relation, and asked for bijective proofs. We give a bijection and prove these results. Using this bijection we also give a new proof that the number of Dyck paths of order $n$ with $k$ peaks is the Narayana number. By double counting super Schröder paths, we also get an identity involving produc…
For a real number $t$, let $r_\ell(t)$ be the total weight of all $t$-large Schröder paths of length $\ell$, and $s_\ell(t)$ be the total weight of all $t$-small Schröder paths of length $\ell$. For constants $α, β$, in this article we derive recurrence formulae for the determinats of the Hankel matrices $\det_{1\le i,j\le n} (αr_{i+j-2}(t) +βr_{i+j-1}(t))$, $\det_{1\le i,j\le n} (αr_{i+j-1}(t) +βr_{i+j}(t))$, $\det_{1\le i,j\le n} (αs_{i+j-2}(t) +βs_{i+j-1}(t))$, and $\det_{1\le i,j\le n} (αs_{i+j-1}(t) +βs_{i+j}(t))$ combinatorially via suitable lattice path models.
We provide a general and unified combinatorial framework for a number of colored partition identities, which include the five, recently proved analytically by B. Berndt, that correspond to the exceptional modular equations of prime degree due to H. Schroeter, R. Russell and S. Ramanujan. Our approach generalizes that of S. Kim, who has given a bijective proof for two of these five identities, namely the ones modulo 7 (also known as the Farkas-Kra identity) and modulo 3. As a consequence of our method, we determine bijective proofs also for the two highly nontrivial identities modulo 5 and 11, …
A permutation $π\in S_n$ is said to {\it avoid} a permutation $σ\in S_k$ whenever $π$ contains no subsequence with all of the same pairwise comparisons as $σ$. For any set $R$ of permutations, we write $S_n(R)$ to denote the set of permutations in $S_n$ which avoid every permutation in $R$. In 1985 Simion and Schmidt showed that $|S_n(132, 213, 123)|$ is equal to the Fibonacci number $F_{n+1}$. In this paper we generalize this result in several ways. We first use a result of Mansour to show that for any permutation $τ$ in a certain infinite family of permutations, $|S_n(132, 213, τ)|$ is given…
Pattern avoidance in alternating sign rectangles I: Extended avoidance
University of Ljubljana, Faculty of Mathematics and Physics（卢布尔雅那大学数学物理学院）；Institute for Mathematics, Physics and Mechanics（数学、物理与力学研究所）；KTH Royal Institute of Technology（瑞典皇家理工学院）
We consider two families of Pascal-like triangles that have all ones on the left side and ones separated by $m-1$ zeros on the right side. The $m=1$ cases are Pascal's triangle and the two families also coincide when $m=2$. Members of the first family obey Pascal's recurrence everywhere inside the triangle. We show that the $m$-th triangle can also be obtained by reversing the elements up to and including the main diagonal in each row of the $(1/(1-x^m),x/(1-x))$ Riordan array. Properties of this family of triangles can be obtained quickly as a result. The $(n,k)$-th entry in the $m$-th member…
