A307030 Number of permutations of [n] with overlapping adjacent runs.

1, 1, 3, 11, 49, 263, 1653, 11877, 95991, 862047, 8516221, 91782159, 1071601285, 13473914281, 181517350571, 2608383775171, 39824825088809, 643813226048935, 10986188094959045, 197337931571468445, 3721889002400665951, 73539326922210382215, 1519081379788242418149, 32743555520207058219615, 735189675389014372317381

Offset: 1

Author

Cyril Banderier, Mar 20 2019

Keywords

• nonn

Comments

The one-line notation of any permutation p has a unique factorization into runs p = B1,B2,...,Bk, where each Bi is a run (a sequence of increasing values), and the first element of B(i+1) is smaller than the last element of Bi. If the permutation p is such that each pair of adjacent runs Bi, B(i+1) have an overlapping range, that is the intervals [min(Bi),max(Bi)] and [min(B(i+1)),max(B(i+1))] have a nonempty intersection, then we say that p has overlapping runs.

a(n) is also the number of permutations of size n transformed by a pop-stack sorting (see Links below).

Examples

			For n = 3 the a(3) = 3 permutations with overlapping runs are 123, 132, 213.
		

Links

• Bjarki Ágúst Guðmundsson, Table of n, a(n) for n = 1..450
• Andrei Asinowski, Cyril Banderier, Sara Billey, Benjamin Hackl and Svante Linusson, Pop-stack sorting and its image: Permutations with overlapping runs (2019), preprint.
• Anders Claesson, Bjarki Ágúst Guðmundsson and Jay Pantone, Counting pop-stacked permutations in polynomial time, arXiv:1908.08910 [math.CO], 2019.
• Colin Defant and Nathan Williams, Semidistrim Lattices, arXiv:2111.08122 [math.CO], 2021.

Crossrefs

Cf. A001855, A008292.

Row sums of A309993.

Extensions

Added more terms (from the Claesson/Guðmundsson/Pantone reference), Joerg Arndt, Aug 26 2019

Definition corrected by Bjarki Ágúst Guðmundsson, Aug 26 2019