A394193 - OEIS

%I #8 Mar 17 2026 19:02:46

%S 192,384,384,288,1920,4800,5120,3840,1536,23040,69120,76800,57600,

%T 27648,15360,322560,1128960,1290240,967680,516096,322560,184320,

%U 5160960,20643840,24084480,18063360,10321920,6881280,4423680,2580480,92897280,418037760,495452160,371589120,222953472,154828800,106168320,69672960,41287680

%N Irregular triangle read by rows: T(n,k) is the number of linear intervals of height k in the weak order on D_n (0 <= k <= n-1).

%C An interval is linear of height k if it is isomorphic to the total order on k+1 elements.

%F Row sums give A394198.

%e Triangle begins:

%e    192,    384,    384,    288;

%e   1920,   4800,   5120,   3840,   1536;

%e  23040,  69120,  76800,  57600,  27648,  15360;

%o (SageMath)

%o def T(n, k):

%o     if k < 0 or k >= n:

%o         return 0

%o     nW = factorial(n) * 2^(n-1)

%o     if k == 0:

%o         return nW

%o     if k == 1:

%o         return nW * n/2

%o     s = 2 * (n-k+1)/(k+1)

%o     if k == 3:

%o         s += 1/2

%o     return s * nW

%Y Cf. A002866, A391308 (type A), A394190 (type B), A394198.

%K nonn,tabf

%O 4,1

%A _Ludovic Schwob_, Mar 12 2026