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A356185
The difference between number of even and number of odd Grassmannian permutations of size n.
8
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
0,6
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.
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.
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)
EXAMPLE
For n=3, 123, 231, 312 are even Grassmann permutations, and 132, 213 are the odd ones. Hence a(3) = 1.
MATHEMATICA
Table[2^Floor[1 + (n - 1)/2] - n, {n, 1, 80}]
CROSSREFS
Bisections give: A005803 (even part), A183155 (odd part).
Sequence in context: A288055 A081233 A050676 * A010372 A374299 A199455
KEYWORD
nonn,easy
AUTHOR
STATUS
approved