A321269 Number of permutations on [n] whose up-down signature has nonnegative partial sums and which have exactly three descents.

0, 0, 0, 0, 0, 0, 604, 7296, 54746, 330068, 1756878, 8641800, 40298572, 180969752, 790697160, 3385019968, 14270283414, 59457742524, 245507935018, 1006678811272, 4105447763032, 16672235476128, 67482738851220, 272439143364672, 1097660274098482, 4415486996246052

Offset: 1

Author

Sam Spiro, Nov 01 2018

Keywords

• nonn
• easy

Comments

Also the number of permutations of [n] of odd order whose M statistic (as defined in the Spiro paper) is equal to three.

Examples

			The permutations counted by a(7) include 1237654 and 17265243.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1660
• S. Spiro, Ballot Permutations, Odd Order Permutations, and a New Permutation Statistic, arXiv preprint arXiv:1810.00993 [math.CO], 2018.
• Index entries for linear recurrences with constant coefficients, signature (24,-260,1684,-7278,22172,-49004,79596,-95065,82508,-50616,20800,-5136,576).

Crossrefs

Cf. A000246, A005803, A321268, A303285, A177042.

Column k=3 of A321280.

Programs

Mathematica

t[n_, k_] := Sum[(-1)^j (k - j)^n Binomial[n + 1, j], {j, 0, k}];
a[n_] := If[n<7, 0, 4 t[n-1, 4] - (Binomial[n, 3] - Binomial[n, 2] + 4) * 2^(n-2) - 22 Binomial[n, 5] + 16 Binomial[n, 4] - 4 Binomial[n, 3] + 2n];
Array[a, 30] (* Jean-François Alcover, Feb 29 2020, from Sam Spiro's 1st formula *)

PARI

concat([0,0,0,0,0,0], Vec(2*x^7*(302 - 3600*x + 18341*x^2 - 52006*x^3 + 89327*x^4 - 94728*x^5 + 61016*x^6 - 23368*x^7 + 5424*x^8 - 576*x^9) / ((1 - x)^6*(1 - 2*x)^4*(1 - 3*x)^2*(1 - 4*x)) + O(x^30))) \\ Colin Barker, Mar 07 2019

Formula

From Sam Spiro, Mar 07 2019: (Start)

a(n) = 4*A008292(n-1,4)-(binomial(n,3)-binomial(n,2)+4)*2^(n-2)-22*binomial(n,5)+16*binomial(n,4)-4*binomial(n,3)+2n for n>3.

a(n) = A065826(n-1,4)-(binomial(n,3)-binomial(n,2)+4)*2^(n-2)-22*binomial(n,5)+16*binomial(n,4)-4*binomial(n,3)+2n for n>3.

a(n) = 4^n-4*n*3^(n-1)+9*binomial(n,2)*2^(n-2)-binomial(n,3)*2^(n-2)-2^n-8*binomial(n,3)-22*binomial(n,5)+16*binomial(n,4)+2*n for n>3.

(End)

From Colin Barker, Mar 07 2019: (Start)

G.f.: 2*x^7*(302 - 3600*x + 18341*x^2 - 52006*x^3 + 89327*x^4 - 94728*x^5 + 61016*x^6 - 23368*x^7 + 5424*x^8 - 576*x^9) / ((1 - x)^6*(1 - 2*x)^4*(1 - 3*x)^2*(1 - 4*x)).

a(n) = 24*a(n-1) - 260*a(n-2) + 1684*a(n-3) - 7278*a(n-4) + 22172*a(n-5) - 49004*a(n-6) + 79596*a(n-7) - 95065*a(n-8) + 82508*a(n-9) - 50616*a(n-10) + 20800*a(n-11) - 5136*a(n-12) + 576*a(n-13) for n>16.

(End)

Extensions

More terms from Alois P. Heinz, Nov 01 2018