Containing All Permutations

Abstract

Numerous versions of the question "what is the shortest object containing all permutations of a given length?" have been asked over the past fifty years: by Karp (via Knuth) in 1972;by Chung, Diaconis, and Graham in 1992;by Ashlock and Tillotson in 1993;and by Arratia in 1999. The large variety of questions of this form, which have previously been considered in isolation, stands in stark contrast to the dearth of answers. We survey and synthesize these questions and their partial answers, introduce infinitely more related questions, and then establish an improved upper bound for one of these questions. c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 128 MICHAEL ENGEN is a Ph.D. candidate at the University of Florida under the supervision of Vincent Vatter. In the fall of 2019, he was a Chateaubriand fellow at Université Paris Nord under the supervision of Frédérique Bassino.

An asymptotically tight lower bound for superpatterns with small alphabets

“…Lastly, we demonstrate how our result contradicts a conjecture by Gupta [7] (see also the second item in the final section of [4]). This conjecture is concerned with "bi-directional circular pattern containment".…”

“…In light of this, one is left to wonder if a revised version of the conjecture from [5] holds true. We answer this in the affirmative by considering a "stricter regime" of the superpattern problem which has received attention recently (see [3,4]).…”

“…. Now, if such a choice/embedding of indices exist, then so will the "greedy embedding" of τ where we take i 1 = min{i : i ∈ σ −1 (τ (1))} and iteratively for j 4 Conversely, if we can construct i 1 , . .…”

“…This is morally because walk(0, τ ) will mimic the greedy embedding of τ , and has infinite cost if and only if the greedy embedding fails. 4 Indeed, suppose i…”

“…As some evidence towards this conjecture, Miller showed that there exist k-superpatterns of length (k 2 + k)/2 (i.e., f (k) ≤ (k 2 + k)/2) [14]. And later Engen and Vatter improved this to show f (k) ≤ (k 2 + 1)/2 [4]. However in forthcoming work [9], the author will show that f (k) ≤ 15 32 k 2 + O(k), refuting the claim that the constant 1/2 is tight.…”