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If L is a partition of n, the rank of L is the size of the largest part minus the number of parts. Under the uniform distribution on partitions, Bringmann, Mahlburg, and Rhoades showed that the rank statistic has a limiting distribution. We identify the limit as the difference between two independent extreme value distributions and as the distribution of B(T) where B(t) is standard Brownian motion and T is the first time that an independent three-dimensional Brownian motion hits the unit sphere. The same limit holds for the crank.
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This note gives a central limit theorem for the length of the longest subsequence of a random permutation which follows some repeating pattern. This includes the case of any fixed pattern of ups and downs which has at least one of each, such as the alternating case considered by Stanley in [2] and Widom in [3]. In every case considered the convergence in the limit of long permutations is to normal with mean and variance linear in the length of the permutations.
Hellys theorem is a classical result concerning the intersection patterns of convex sets in $mathbb{R}^d$. Two important generalizations are the colorful version and the fractional version. Recently, B{a}r{a}ny et al. combined the two, obtaining a colorful fractional Helly theorem. In this paper, we give an improved version of their result.
A generalized crank ($k$-crank) for $k$-colored partitions is introduced. Following the work of Andrews-Lewis and Ji-Zhao, we derive two results for this newly defined $k$-crank. Namely, we first obtain some inequalities between the $k$-crank counts $M_{k}(r,m,n)$ for $m=2,3$ and $4$, then we prove the positivity of symmetrized even $k$-crank moments weighted by the parity for $k=2$ and $3$. We conclude with several remarks on furthering the study initiated here.
Let $N(leq m,n)$ denote the number of partitions of $n$ with rank not greater than $m$, and let $M(leq m,n)$ denote the number of partitions of $n$ with crank not greater than $m$. Bringmann and Mahlburg observed that $N(leq m,n)leq M(leq m,n)leq N(leq m+1,n)$ for $m<0$ and $1leq nleq 100$. They also pointed out that these inequalities can be restated as the existence of a re-ordering $tau_n$ on the set of partitions of $n$ such that $|text{crank}(lambda)|-|text{rank}(tau_n(lambda))|=0$ or $1$ for all partitions $lambda$ of $n$, that is, the rank and the crank are nearly equal distributions over partitions of $n$. In the study of the spt-function, Andrews, Dyson and Rhoades proposed a conjecture on the unimodality of the spt-crank, and they showed that this conjecture is equivalent to the inequality $N(leq m,n)leq M(leq m,n)$ for $m<0$ and $ngeq 1$. We proved this conjecture by combiantorial arguments. In this paper, we prove the inequality $N(leq m,n)leq M(leq m,n)$ for $m<0$ and $ngeq 1$. Furthermore, we define a re-ordering $tau_n$ of the partitions $lambda$ of $n$ and show that this re-ordering $tau_n$ leads to the nearly equal distribution of the rank and the crank. Using the re-ordering $tau_n$, we give a new combinatorial interpretation of the function ospt$(n)$ defined by Andrews, Chan and Kim, which immediately leads to an upper bound for $ospt(n)$ due to Chan and Mao.
The Hales-Jewett theorem for alphabet of size 3 states that whenever the Hales-Jewett cube [3]^n is r-coloured there is a monochromatic line (for n large). Conlon and Kamcev conjectured that, for any n, there is a 2-colouring of [3]^n for which there is no monochromatic line whose active coordinate set is an interval. In this note we disprove this conjecture.
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