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We exhibit a bijection between recently-introduced combinatorial objects known as valid hook configurations and certain weighted set partitions. When restricting our attention to set partitions that are matchings, we obtain three new combinatorial interpretations of Lassalle's sequence. One of these interpretations involves permutations that have exactly one preimage under the (West) stack-sorting map. We prove that the sequences obtained by counting these permutations according to their first entries are symmetric, and we conjecture that they are log-concave. We also obtain new recurrence relations involving Lassalle's sequence and the sequence that enumerates valid hook configurations. We end with several suggestions for future work.

Korpelainen, Lozin, and Razgon conjectured that a hereditary property of graphs which is well-quasi-ordered by the induced subgraph order and defined by only finitely many minimal forbidden induced subgraphs is labelled well-quasi-ordered, a notion stronger than that of n-well-quasiorder introduced by Pouzet in the 1970s. We present a counterexample to this conjecture. In fact, we exhibit a hereditary property of graphs which is well-quasi-ordered by the induced subgraph order and defined by finitely many minimal forbidden induced subgraphs yet is not 2-wellquasi-ordered. This counterexample is based on the widdershins spiral, which has received some study in the area of permutation patterns.

We establish an exact formula for the length of the shortest permutation containing all layered permutations of length n, proving a conjecture of Gray.We establish the following result, which gives an exact formula for the length of the shortest nuniversal permutation for the class of layered permutations, verifying a conjecture of Gray [3]. Definitions follow the statement.Theorem 1. For all n, the length of the shortest permutation that is n-universal for the layered permutations is given by the sequence defined by apnq " n`mintapkq`apn´k´1q : 0 ď k ď n´1u (:) and ap0q " 0.Up to shifting indices by 1, the sequence apnq in Theorem 1 is sequence A001855 in the OEIS [6]. It seems to have first appeared in Knuth's The Art of Computer Programming, Volume 3 [4, Section 5.3.1, Eq.(3)], where it is related to sorting by binary insertion. Knuth shows there that (in our indexing conventions), apnq " pn`1qrlog 2 pn`1qs´2 rlog 2 pn`1qs`1 .This formula also shows that the minimum in (:) is attained when k " tn{2u. We refer the reader to the OEIS for further information about this old and interesting sequence.

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.