A356185 The difference between number of even and number of odd Grassmannian permutations of size n.

1, 1, 0, 1, 0, 3, 2, 9, 8, 23, 22, 53, 52, 115, 114, 241, 240, 495, 494, 1005, 1004, 2027, 2026, 4073, 4072, 8167, 8166, 16357, 16356, 32739, 32738, 65505, 65504, 131039, 131038, 262109, 262108, 524251, 524250, 1048537, 1048536, 2097111, 2097110, 4194261, 4194260

Offset: 0

Author

Per W. Alexandersson, Jul 28 2022

Keywords

• nonn
• easy

Comments

A permutation is Grassmann if it has at most one descent. A closed-form formula was proved by J. B. Gil and J. A. Tomasko.

Examples

			For n=3, 123, 231, 312 are even Grassmann permutations, and 132, 213 are the odd ones. Hence a(3) = 1.
		

Links

• Juan B. Gil and Jessica A. Tomasko, Restricted Grassmannian permutations, arXiv:2112.03338 [math.CO], 2021.
• Juan B. Gil and Jessica A. Tomasko, Restricted Grassmannian permutations, Enum. Combin. Appl. 2 (2022), no. 4, Article #S4PP6.
• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,2).

Crossrefs

Cf. A000325, A001710, A032085, A060546, A122746, A233411.

Bisections give: A005803 (even part), A183155 (odd part).

Programs

Mathematica

Table[2^Floor[1 + (n - 1)/2] - n, {n, 1, 80}]

Formula

a(n) = 2^(1+floor((n-1)/2))-n.

From Alois P. Heinz, Jul 28 2022: (Start)

G.f.: -(4*x^3-3*x^2-x+1)/((2*x^2-1)*(x-1)^2).

a(n) = A000325(n) - A233411(n) = A060546(n) - n = 2^ceiling(n/2) - n.

a(n) = A000325(n) - 2*A032085(n) = A000325(n) - 2*A122746(n-2) for n>=2. (End)