A260506 Number of signed permutations of length n that are sortable to the identity permutation by some sequence of cdr (context-directed reversal) and cds (context-directed swap) moves.

1, 3, 18, 140, 1384, 16428, 228248

Offset: 1

Author

Caleb Stanford, Jul 27 2015

Keywords

• nonn
• more

Comments

a(n) is equivalently the number of signed permutations such that in the reality-desire graph of the permutation, 0 and n are in different cycles.

It is also the number of signed permutations such that the overlap of that permutation contains a partition into two vertex sets, V_1 and V_2, such that (i) vertices (0,1) and (n,n+1) are in different cycles, and (ii) every vertex in V_1 is adjacent to an even number of vertices in V_2, and every vertex in V_2 is adjacent to an even number of vertices in V_1.

It is expected (but not proven) asymptotically that a(n) is about one-third of all permutations, i.e., it is about (2^n * n!) / 3.

Examples

			a(2) = 3 because out of the 8 signed permutations of size 2, only [1,2], [1,-2], and [-1,2] are sortable. They each take one cdr move. Three other permutations, [-2,-1], [2,-1], and [-2,1] are not sortable to the identity [1,2] but instead sort to [-2,-1]. Finally, [2,1] and [-1,-2] sort to neither [1,2] nor [-2,-1].
		

Links

• K. L. M. Adamyk, E. Holmes, G. R. Mayfield, D. J. Moritz, M. Scheepers, B. E. Tenner, and H. C. Wauck, Sorting permutations: games, genomes, and cycles, arXiv:1410.2353 [math.CO], 2014.

Crossrefs

A249165 counts the special case of this property for unsigned permutations. Note that for unsigned permutations, there are no valid cdr moves, so sortability by cdr and cds is equivalent to sortability by cds only.

A000165 counts the total number of signed permutations.