A363236 Number of permutations p of [n] such that each element in p has at least one neighbor with opposite parity.

1, 0, 2, 2, 16, 36, 288, 1152, 10368, 57600, 604800, 4320000, 51840000, 453600000, 6147187200, 63605606400, 962415820800, 11500218777600, 192255565824000, 2605984690176000, 47721518530560000, 723526168780800000, 14407079038894080000, 241602987041095680000

Offset: 0

Author

Alois P. Heinz, May 22 2023

Keywords

• nonn

Examples

			a(0) = 1: (), the empty permutation.
a(1) = 0.
a(2) = 2: 12, 21.
a(3) = 2: 123, 321.
a(4) = 16: 1234, 1243, 1423, 1432, 2134, 2143, 2314, 2341, 3214, 3241, 3412, 3421, 4123, 4132, 4312, 4321.
a(5) = 36: 12345, 12354, 12534, 12543, 14325, 14352, 14523, 14532, 21345, 21543, 23145, 23541, 25143, 25341, 32145, 32154, 32514, 32541, 34125, 34152, 34512, 34521, 41325, 41523, 43125, 43521, 45123, 45321, 52134, 52143, 52314, 52341, 54123, 54132, 54312, 54321.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..465
• Wikipedia, Permutation

Crossrefs

Cf. A001622, A092186, A363180, A363181.

Formula

a(n) ~ phi^n * n! / (5^(1/4) * 2^(n-1)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, May 26 2023