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A344216

a(n) = n!*((n+1)/2 + 2*Sum_{k=2..n-1}(n-k)/(k+1)).

1, 3, 16, 104, 768, 6336, 57888, 581472, 6379200, 75977280, 977045760, 13499930880, 199537067520, 3142504512000, 52546707763200, 929908914278400, 17366044153651200, 341336836618444800, 7044417438363648000

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OFFSET

1,2

COMMENTS

Conjecture: a(n) is the number of linear intervals in the weak order on the symmetric group S_n. An interval is linear if it is isomorphic to a total order.

The above conjecture has been proved by Chenevière in his Ph. D. thesis (see Links). - Ludovic Schwob, Dec 31 2025

LINKS

Table of n, a(n) for n=1..19.

Clément Chenevière, Enumerative study of intervals in lattices of Tamari type, Ph. D. thesis, Univ. Strasbourg (France), Ruhr-Univ. Bochum (Germany), HAL tel-04255439 [math.CO], 2024. See p. 145.

FORMULA

From Peter Luschny, May 13 2021: (Start)

a(n) = (1/2) * n! * (4 * (n + 1) * H(n) - 9*n + 3), where H(n) are the harmonic numbers H(n) = A001008(n)/A002805(n).

a(n) = n! * [x^n] (3 - 8*x - 4*log(1 - x))/(2*(x - 1)^2).

a(n) = ((2*n^2 - 5*n - 1)*a(n-1) - (n^3 - 3*n^2 + 2*n)*a(n-2))/(n - 3) for n >= 4. (End)

EXAMPLE

For S_3, among the 17 intervals in the hexagon-shaped lattice, only the full lattice is not linear.

MAPLE

a := n -> (1/2)*n!*(4*(n + 1)*harmonic(n) - 9*n + 3):

# Or:

egf := (3 - 8*x - 4*ln(1 - x))/(2*(x - 1)^2):

ser := series(egf, x, 24): a := n -> n!*coeff(ser, x, n):

seq(a(n), n=1..19); # Peter Luschny, May 13 2021

MATHEMATICA

Join[{1}, RecurrenceTable[{(n - 3) a[n] == (2 n^2 - 5 n - 1) a[n - 1] - (n^3 - 3 n^2 + 2 n) a[n - 2], a[2] == 3, a[3] == 16}, a, {n, 2, 19}]] (* Peter Luschny, May 13 2021 *)

PROG

(SageMath)

def a(n):

return factorial(n)*((n+1)/2+2*sum((n-k)/(k+1) for k in range(2, n)))

(PARI) a(n) = n!*((n+1)/2+2*sum(k=2, n-1, (n-k)/(k+1))); \\ Michel Marcus, May 13 2021

CROSSREFS

Cf. A344136, A344191, A344228 for similar sequences.

Cf. A007767 for all intervals in the weak order on S_n.

Cf. A001008, A002805.

Cf. A391308 for linear intervals distinguished by height.

Sequence in context: A074542 A366050 A368934 * A393734 A105622 A110903

Adjacent sequences: A344213 A344214 A344215 * A344217 A344218 A344219