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A321269 Number of permutations on [n] whose up-down signature has nonnegative partial sums and which have exactly three descents. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

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

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).

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)

EXAMPLE

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

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-Fran├žois Alcover, Feb 29 2020, from Sam Spiro's 1st formula *)

PROG

(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

CROSSREFS

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

Column k=3 of A321280.

Sequence in context: A031792 A020383 A234109 * A234343 A234338 A250969

Adjacent sequences:  A321266 A321267 A321268 * A321270 A321271 A321272

KEYWORD

nonn,easy

AUTHOR

Sam Spiro, Nov 01 2018

EXTENSIONS

More terms from Alois P. Heinz, Nov 01 2018

STATUS

approved

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Last modified November 26 21:07 EST 2021. Contains 349344 sequences. (Running on oeis4.)