The reverse mathematics of non-decreasing subsequences

The reverse mathematics of non-decreasing subsequences

Every function over the natural numbers has an infinite subdomain on which the function is non-decreasing. Motivated by a question of Dzhafarov and Schweber, we study the reverse mathematics of variants of this statement. It turns out that this statement restricted to computably bounded functions is computationally weak and does not imply the existence of the halting set. On the other hand, we ...

The Surprising Mathematics of Longest Increasing Subsequences

Partitioning Permutations into Increasing and Decreasing Subsequences

Increasing and Decreasing Subsequences and Their Variants

We survey the theory of increasing and decreasing subsequences of permutations. Enumeration problems in this area are closely related to the RSK algorithm. The asymptotic behavior of the expected value of the length is(w) of the longest increasing subsequence of a permutation w of 1, 2, . . . , n was obtained by Vershik-Kerov and (almost) by Logan-Shepp. The entire limiting distribution of is(w...

Reverse-engineering Reverse Mathematics

An important open problem in Reverse Mathematics ([16, 25]) is the reduction of the first-order strength of the base theory from IΣ1 to I∆0 + exp. The system ERNA, a version of Nonstandard Analysis based on the system I∆0 + exp, provides a partial solution to this problem. Indeed, Weak König’s lemma and many of its equivalent formulations from Reverse Mathematics can be ‘pushed down’ into ERNA,...