2 research outputs found
The variance and the asymptotic distribution of the length of longest k-alternating subsequences
We obtain an explicit formula for the variance of the number of k-peaks in a uniformly random permutation. This is then used to obtain an asymptotic formula for the variance of the length of longest k-alternating subsequence in random permutations. Also a central limit is proved for the latter statistic
Exponential Erd\H{o}s-Szekeres theorem for matrices
In 1993, Fishburn and Graham established the following qualitative extension
of the classical Erd\H{o}s-Szekeres theorem. If is sufficiently large with
respect to , then any real matrix contains an
submatrix in which every row and every column is monotone. We prove that the
smallest such is at most , greatly improving the previously
best known double-exponential upper bound, and getting close to the best known
lower bound .
In particular, we prove the following surprising sharp transition in the
asymmetric setting. On one hand, every matrix
contains an submatrix, in which every row is mononote. On the other
hand, there exist matrices containing no
such submatrix .Comment: 10 pages, 1 figur