login
Triangle of allowable Stirling numbers of the second kind a(n,k).
2

%I #28 Feb 10 2020 10:41:42

%S 1,1,1,1,2,1,1,3,4,1,1,4,11,6,1,1,5,26,23,9,1,1,6,57,72,50,12,1,1,7,

%T 120,201,222,86,16,1,1,8,247,522,867,480,150,20,1,1,9,502,1291,3123,

%U 2307,1080,230,25,1,1,10,1013,3084,10660,10044,6627,2000,355,30,1

%N Triangle of allowable Stirling numbers of the second kind a(n,k).

%C Row sums = A007476 starting (1, 2, 4, 9, 23, 65, 199, 654, 2296, 8569, ...).

%C a(n,k) counts restricted growth words of length n in the letters {1, ..., k} where every even entry appears exactly once.

%H Yue Cai and Margaret Readdy, <a href="http://arxiv.org/abs/1506.03249">Negative q-Stirling numbers</a>, arXiv:1506.03249 [math.CO], 2015.

%F a(n,k) = a(n-1,k-1) + ceiling(k/2)*a(n-1,k) for n >= 1 and 1 <= k <= n with boundary conditions a(n,0) = KroneckerDelta[n,0].

%F a(n,2) = n-1.

%F a(n,n-1) = floor(n/2)*ceiling(n/2).

%e a(4,1) = 1 via 1111;

%e a(4,2) = 3 via 1211, 1121, 1112;

%e a(4,3) = 4 via 1213, 1231, 1233, 1123;

%e a(4,4) = 1 via 1234.

%e Triangle starts:

%e 1;

%e 1, 1;

%e 1, 2, 1;

%e 1, 3, 4, 1;

%e 1, 4, 11, 6, 1;

%e ...

%t a[_, 1] = a[n_, n_] = 1;

%t a[n_, k_] := a[n, k] = a[n-1, k-1] + Ceiling[k/2] a[n-1, k];

%t Table[a[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Dec 15 2018 *)

%Y Cf. A007476 (row sums), A246118 (essentially the same triangle).

%K nonn,tabl

%O 1,5

%A _Margaret A. Readdy_, Mar 16 2015