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Register a new userMore bijections for Entringer and Arnold families
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The Euler number $E_n$ (resp. Entringer number $E_{n,k}$) enumerates the alternating (down-up) permutations of ${1,dots,n}$ (resp. starting with $k$). The Springer number $S_n$ (resp. Arnold number $S_{n,k}$) enumerates the type $B$ alternating permutations (resp. starting with $k$). In this paper, using bijections we first derive the counterparts in {em Andre permutations} and {em Simsun permutations} for the Entringer numbers $(E_{n,k})$, and then the counterparts in {em signed Andre permutations} and {em type $B$ increasing 1-2 trees} for the Arnold numbers $(S_{n,k})$.
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The class of ranked tree-child networks, tree-child networks arising from an evolution process with a fixed embedding into the plane, has recently been introduced by Bienvenu, Lambert, and Steel. These authors derived counting results for this class. In this note, we will give bijective proofs of three of their results. Two of our bijections answer questions raised in their paper.
The extremal problems regarding the maximum possible size of intersecting families of various combinatorial objects have been extensively studied. In this paper, we investigate supersaturation extensions, which in this context ask for the minimum number of disjoint pairs that must appear in families larger than the extremal threshold. We study the minimum number of disjoint pairs in families of permutations and in $k$-uniform set families, and determine the structure of the optimal families. Our main tool is a removal lemma for disjoint pairs. We also determine the typical structure of $k$-uniform set families without matchings of size $s$ when $n ge 2sk + 38s^4$, and show that almost all $k$-uniform intersecting families on vertex set $[n]$ are trivial when $nge (2+o(1))k$.
Johnson recently proved Armstrongs conjecture which states that the average size of an $(a,b)$-core partition is $(a+b+1)(a-1)(b-1)/24$. He used various coordinate changes and one-to-one correspondences that are useful for counting problems about simultaneous core partitions. We give an expression for the number of $(b_1,b_2,cdots, b_n)$-core partitions where ${b_1,b_2,cdots,b_n}$ contains at least one pair of relatively prime numbers. We also evaluate the largest size of a self-conjugate $(s,s+1,s+2)$-core partition.
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.
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.
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