Search Results

For a permutation $\sigma \in S_n$, let $\ell(\sigma)$ denote the maximal length of an increasing subsequence in $\sigma$. Define $$ \ell_n = \frac{1}{n!} \sum_{\sigma \in S_n} \ell(\sigma), $$ the av …

For a Young diagram $\lambda$ of size $n$, let $f^\lambda$ denote the number of standard Young tableaux of shape $\lambda$ (discussed above in Mark Wildon's answer about the hook length formula). Then …

I'm confused about why no one has mentioned Stirling's formula for the factorial function $n!$, clearly the most famous and important formula in asymptotic combinatorics, and easily one of the most im …

The Rogers-Ramanujan identities are partition identities, i.e., statements that equate the number of integer partitions of an integer $n$ belonging to two different partition classes. There are two id …

@user61318 and @JosephORourke, thanks for the advertisement for my book. @chubakueono, as far as I know the answer to your questions are no and no. Pages 148-149 in my book have the state of the art …

Another large family of distributions that contains both $U[0,1]$ inputs (leading to the usual Plancherel measure) and $U\{1,2,\ldots,d\}$ (as in Ryan O'Donnell's answer) as special cases is the follo …

A Gaussian limiting distribution is possible (if you allow a scaling operation to bring the mean and variance down to $O(1)$; this was not specified in the question but seems like a fair assumption). …

TL;DR: I have a proof of your conjectured asymptotic formula, modulo the correctness of a certain alternative description of your $c_n$ sequence. I tried to complete Iosif Pinelis's elegant analysis …