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Register a new userJohnsons bijections and their application to counting simultaneous core partitions
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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.
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For $n$ and $k$ integers we introduce the notion of some partition of $n$ being able to generate another partition of $n$. We solve the problem of finding the minimum size partition for which the set of partitions this partition can generate contains all size-$k$ partitions of $n$. We describe how this result can be applied to solving a class of combinatorial optimization problems.
We study a curious class of partitions, the parts of which obey an exceedingly strict congruence condition we refer to as sequential congruence: the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part congruent to zero modulo the length of the partition. It turns out these obscure-seeming objects are embedded in a natural way in partition theory. We show that sequentially congruent partitions with largest part $n$ are in bijection with the partitions of $n$. Moreover, we show sequentially congruent partitions induce a bijection between partitions of $n$ and partitions of length $n$ whose parts obey a strict frequency congruence condition -- the frequency (or multiplicity) of each part is divisible by that part -- and prove families of similar bijections, connecting with G. E. Andrewss theory of partition ideals.
For non-negative integers $n$ and $k$ with $n ge k$, a {em $k$-minor} of a partition $lambda = [lambda_1, lambda_2, dots]$ of $n$ is a partition $mu = [mu_1, mu_2, dots]$ of $n-k$ such that $mu_i le lambda_i$ for all $i$. The multiset $widehat{M}_k(lambda)$ of $k$-minors of $lambda$ is defined as the multiset of $k$-minors $mu$ with multiplicity of $mu$ equal to the number of standard Young tableaux of skew shape $lambda / mu$. We show that there exists a function $G(n)$ such that the partitions of $n$ can be reconstructed from their multisets of $k$-minors if and only if $k le G(n)$. Furthermore, we prove that $lim_{n rightarrow infty} G(n)/n = 1$ with $n-G(n) = O(n/log n)$. As a direct consequence of this result, the irreducible representations of the symmetric group $S_n$ can be reconstructed from their restrictions to $S_{n-k}$ if and only if $k le G(n)$ for the same function $G(n)$. For a minor $mu$ of the partition $lambda$, we study the excitation factor $E_mu (lambda)$, which appears as a crucial part in Naruses Skew-Shape Hook Length Formula. We observe that certain excitation factors of $lambda$ can be expressed as a $mathbb{Q}[k]$-linear combination of the elementary symmetric polynomials of the hook lengths in the first row of $lambda$ where $k = lambda_1$ is the number of cells in the first row of $lambda$.
We find a formula for the number of permutations of $[n]$ that have exactly $s$ runs up and down. The formula is at once terminating, asymptotic, and exact.
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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