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Register a new userCombinatorial bijections from hatted avoiding permutations in $S_n(132)$ to generalized Dyck and Motzkin paths
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We introduce a new concept of permutation avoidance pattern called hatted pattern, which is a natural generalization of the barred pattern. We show the growth rate of the class of permutations avoiding a hatted pattern in comparison to barred pattern. We prove that Dyck paths with no peak at height $p$, Dyck paths with no $ud... du$ and Motzkin paths are counted by hatted pattern avoiding permutations in $s_n(132)$ by showing explicit bijections. As a result, a new direct bijection between Motzkin paths and permutations in $s_n(132)$ without two consecutive adjacent numbers is given. These permutations are also represented on the Motzkin generating tree based on the Enumerative Combinatorial Object (ECO) method.
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A {em k-generalized Dyck path} of length $n$ is a lattice path from $(0,0)$ to $(n,0)$ in the plane integer lattice $mathbb{Z}timesmathbb{Z}$ consisting of horizontal-steps $(k, 0)$ for a given integer $kgeq 0$, up-steps $(1,1)$, and down-steps $(1,-1)$, which never passes below the x-axis. The present paper studies three kinds of statistics on $k$-generalized Dyck paths: number of $u$-segments, number of internal $u$-segments and number of $(u,h)$-segments. The Lagrange inversion formula is used to represent the generating function for the number of $k$-generalized Dyck paths according to the statistics as a sum of the partial Bell polynomials or the potential polynomials. Many important special cases are considered leading to several surprising observations. Moreover, enumeration results related to $u$-segments and $(u,h)$-segments are also established, which produce many new combinatorial identities, and specially, two new expressions for Catalan numbers.
A combinatorial structure, $mathcal{F}$, with counting sequence ${a_n}_{nge 0}$ and ordinary generating function $G_mathcal{F}=sum_{nge0} a_n x^n$, is positive algebraic if $G_mathcal{F}$ satisfies a polynomial equation $G_mathcal{F}=sum_{k=0}^N p_k(x),G_mathcal{F}^k $ and $p_k(x)$ is a polynomial in $x$ with non-negative integer coefficients. We show that every such family is associated with a normed $mathbf{n}$-magma. An $mathbf{n}$-magma with $mathbf{n}=(n_1,dots, n_k)$ is a pair $mathcal{M}$ and $mathcal{F}$ where $mathcal{M}$ is a set of combinatorial structures and $mathcal{F}$ is a tuple of $n_i$-ary maps $f_i,:,mathcal{M}^{n_i}to mathcal{M}$. A norm is a super-additive size map $||cdot||,:, mathcal{M}to mathbb{N} $. If the normed $mathbf{n}$-magma is free then we show there exists a recursive, norm preserving, universal bijection between all positive algebraic families $mathcal{F}_i$ with the same counting sequence. A free $mathbf{n}$-magma is defined using a universal mapping principle. We state a theorem which provides a combinatorial method of proving if a particular $mathbf{n}$-magma is free. We illustrate this by defining several $mathbf{n}$-magmas: eleven $(1,1)$-magmas (the Fibonacci families), seventeen $(1,2)$-magmas (nine Motzkin and eight Schroder families) and seven $(3)$-magmas (the Fuss-Catalan families). We prove they are all free and hence obtain a universal bijection for each $mathbf{n}$. We also show how the $mathbf{n}$-magma structure manifests as an embedded bijection.
We show bijectively that Dyck paths with all peaks at odd height are counted by the Motzkin numbers and Dyck paths with all peaks at even height are counted by the Riordan numbers.
For any integer $mge 2$ and a set $Vsubset {1,dots,m}$, let $(m,V)$ denote the union of congruence classes of the elements in $V$ modulo $m$. We study the Hankel determinants for the number of Dyck paths with peaks avoiding the heights in the set $(m,V)$. For any set $V$ of even elements of an even modulo $m$, we give an explicit description of the sequence of Hankel determinants in terms of subsequences of arithmetic progression of integers. There are numerous instances for varied $(m,V)$ with periodic sequences of Hankel determinants. We present a sufficient condition for the set $(m,V)$ such that the sequence of Hankel determinants is periodic, including even and odd modulus $m$.
A {em Motzkin path} of length $n$ is a lattice path from $(0,0)$ to $(n,0)$ in the plane integer lattice $mathbb{Z}timesmathbb{Z}$ consisting of horizontal-steps $(1, 0)$, up-steps $(1,1)$, and down-steps $(1,-1)$, which never passes below the x-axis. A {em $u$-segment {rm (resp.} $h$-segment {rm)}} of a Motzkin path is a maximum sequence of consecutive up-steps ({rm resp.} horizontal-steps). The present paper studies two kinds of statistics on Motzkin paths: number of $u$-segments and number of $h$-segments. The Lagrange inversion formula is utilized to represent the weighted generating function for the number of Motzkin paths according to the statistics as a sum of the partial Bell polynomials or the potential polynomials. As an application, a general framework for studying compositions are also provided.
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