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

%I A356185 #30 Jul 30 2022 11:44:35
%S A356185 1,1,0,1,0,3,2,9,8,23,22,53,52,115,114,241,240,495,494,1005,1004,2027,
%T A356185 2026,4073,4072,8167,8166,16357,16356,32739,32738,65505,65504,131039,
%U A356185 131038,262109,262108,524251,524250,1048537,1048536,2097111,2097110,4194261,4194260
%N A356185 The difference between number of even and number of odd Grassmannian permutations of size n.
%C A356185 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.
%H A356185 Juan B. Gil and Jessica A. Tomasko, <a href="https://arxiv.org/abs/2112.03338">Restricted Grassmannian permutations</a>, arXiv:2112.03338 [math.CO], 2021.
%H A356185 Juan B. Gil and Jessica A. Tomasko, <a href="https://doi.org/10.54550/ECA2022V2S4PP6">Restricted Grassmannian permutations</a>, Enum. Combin. Appl. 2 (2022), no. 4, Article #S4PP6.
%H A356185 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,2).
%F A356185 a(n) = 2^(1+floor((n-1)/2))-n.
%F A356185 From _Alois P. Heinz_, Jul 28 2022: (Start)
%F A356185 G.f.: -(4*x^3-3*x^2-x+1)/((2*x^2-1)*(x-1)^2).
%F A356185 a(n) = A000325(n) - A233411(n) = A060546(n) - n = 2^ceiling(n/2) - n.
%F A356185 a(n) = A000325(n) - 2*A032085(n) = A000325(n) - 2*A122746(n-2) for n>=2. (End)
%e A356185 For n=3, 123, 231, 312 are even Grassmann permutations, and 132, 213 are the odd ones. Hence a(3) = 1.
%t A356185 Table[2^Floor[1 + (n - 1)/2] - n, {n, 1, 80}]
%Y A356185 Cf. A000325, A001710, A032085, A060546, A122746, A233411.
%Y A356185 Bisections give: A005803 (even part), A183155 (odd part).
%K A356185 nonn,easy
%O A356185 0,6
%A A356185 _Per W. Alexandersson_, Jul 28 2022