A344216 - OEIS

%I #35 Jan 17 2024 10:34:42

%S 1,3,16,104,768,6336,57888,581472,6379200,75977280,977045760,

%T 13499930880,199537067520,3142504512000,52546707763200,

%U 929908914278400,17366044153651200,341336836618444800,7044417438363648000

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

%C 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 conjecture has been checked up to a(7) = 57888.

%H Clément Chenevière, <a href="https://theses.hal.science/tel-04255439">Enumerative study of intervals in lattices of Tamari type</a>, Ph. D. thesis, Univ. Strasbourg (France), Ruhr-Univ. Bochum (Germany), HAL tel-04255439 [math.CO], 2024. See p. 145.

%F From _Peter Luschny_, May 13 2021: (Start)

%F 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).

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

%F 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)

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

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

%p # Or:

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

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

%p seq(a(n), n=1..19); # _Peter Luschny_, May 13 2021

%t 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 *)

%o (Sage)

%o def a(n):

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

%o (PARI) a(n) = n!*((n+1)/2+2*sum(k=2, n-1, (n-k)/(k+1))); \\ _Michel Marcus_, May 13 2021

%Y Cf. A344136, A344191, A344228 for similar sequences.

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

%Y Cf. A001008, A002805.

%K nonn

%O 1,2

%A _F. Chapoton_, May 13 2021