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A321268 Number of permutations on [n] whose up-down signature has nonnegative partial sums and which have exactly two descents. 3
0, 0, 0, 0, 22, 172, 856, 3488, 12746, 43628, 143244, 457536, 1434318, 4438540, 13611136, 41473216, 125797010, 380341580, 1147318004, 3455325600, 10394291094, 31242645420, 93853769320, 281825553760, 846030314842, 2539248578732, 7620161662556, 22865518160768 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

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

LINKS

Sam Spiro, Table of n, a(n) for n = 1..100

S. Spiro, Ballot Permutations, Odd Order Permutations, and a New Permutation Statistic, arXiv:1810.00993 [math.CO], 2018.

Index entries for linear recurrences with constant coefficients, signature (11,-50,122,-173,143,-64,12).

FORMULA

a(n) = 3*A008292(n-1,3)- 2*binomial(n,3)+binomial(n,2)-1 for n > 1.

a(n) =   A065826(n-1,3)- 2*binomial(n,3)+binomial(n,2)-1 for n > 1.

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

From Colin Barker, Mar 07 2019: (Start)

G.f.: 2*x^5*(11 - 35*x + 32*x^2 - 6*x^3) / ((1 - x)^4*(1 - 2*x)^2*(1 - 3*x)).

a(n) = 11*a(n-1) - 50*a(n-2) + 122*a(n-3) - 173*a(n-4) + 143*a(n-5) - 64*a(n-6) + 12*a(n-7) for n>8.

a(n) = -1 + 3^n - (16+9*2^n)*n/6 + 3*n^2 - n^3/3 for n>1.

(End)

EXAMPLE

Some permutations counted by a(5) include 14253 and 34521.

MATHEMATICA

a[1] = 0; a[n_] := 2n^2 - 2n - 1 - n 2^(n-1) - 2 Binomial[n, 3] + Sum[ Binomial[n, k] (2^k - 2k), {k, 0, n}];

Table[a[n], {n, 1, 28}] (* Jean-Fran├žois Alcover, Nov 11 2018 *)

PROG

(PARI) a(n)={if(n<2, 0, 2*n^2 - 2*n - 1 - n*2^(n-1) - 2*binomial(n, 3) + sum(k=0, n, binomial(n, k)*(2^k - 2*k)))} \\ Andrew Howroyd, Nov 01 2018

(PARI) concat([0, 0, 0, 0], Vec(2*x^5*(11 - 35*x + 32*x^2 - 6*x^3) / ((1 - x)^4*(1 - 2*x)^2*(1 - 3*x)) + O(x^40))) \\ Colin Barker, Mar 07 2019

CROSSREFS

Cf. A000246, A005803, A321269, A303285, A177042.

Column k=2 of A321280.

Sequence in context: A223775 A224407 A243143 * A086604 A041932 A125359

Adjacent sequences:  A321265 A321266 A321267 * A321269 A321270 A321271

KEYWORD

nonn,easy

AUTHOR

Sam Spiro, Nov 01 2018

STATUS

approved

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Last modified January 19 11:16 EST 2022. Contains 350465 sequences. (Running on oeis4.)