A182216 Number of permutations sortable by a double-ended queue.

1, 1, 2, 6, 24, 116, 634, 3762, 23638, 154816, 1046010, 7239440, 51069582, 365879686, 2654987356, 19473381290, 144138193538, 1075285161294, 8076634643892, 61028985689976, 463596673890280, 3538275218777642

Offset: 0

Author

Paul Zimmermann, Apr 19 2012

Keywords

• nonn

Examples

			Up to n=4 all permutations can be sorted by a double-ended queue (deque for short).
For n=5 the permutation 24351 cannot: you first queue 2 on either side, then 4 on either side, then 3 has to be queued on the same side as 4 otherwise it will "block" 2 between 3 and 4, but then whatever side you queue 5, you will block either 2 (between 4 and 5) or 3 (between 4 and 5).
		

Links

• Daniel Denton and Peter Doyle, Table of n, a(n) for n = 0..100
• Michael Albert, Mike Atkinson, Steve Linton, Permutations generated by stacks and deques, Annals of Combinatorics, 14 (2010) 3-16. (gives method of computation).
• Daniel Denton, Methods of computing deque sortable permutations given complete and incomplete information, arXiv:1208.1532.
• Stoyan Dimitrov, Sorting by shuffling methods and a queue, arXiv:2103.04332 [math.CO], 2021.
• Andrew Elvey-Price and Anthony J. Guttmann, Permutations sortable by deques and by two stacks in parallel, arxiv:1508.02273, 2015
• Philippe Flajolet, Bruno Salvy and Paul Zimmermann, Lambda-Upsilon-Omega, The 1989 CookBook, page 105.
• Paul Zimmermann, C program for A182216
• P. Zimmermann, Introduction to Automatic Analysis, 2012.

Crossrefs

Cf. A215257, A216040, A215252.

Extensions

a(13)-a(14) confirmed by Michael Albert, Apr 19 2012

a(16)-a(21) from Michael Albert, Jun 27 2012

a(0)=1 added by N. J. A. Sloane, Sep 12 2012