A177254 - OEIS

%I #8 May 12 2024 02:00:06

%S 1,0,1,0,1,1,0,2,2,1,1,4,6,3,1,5,13,17,12,4,1,21,51,61,44,20,5,1,91,

%T 219,255,185,90,30,6,1,422,1019,1182,867,440,160,42,7,1,2103,5108,

%U 5964,4430,2322,896,259,56,8,1,11226,27448,32373,24406,13118,5292,1638,392,72,9,1

%N Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} having k adjacent blocks (0 <= k <= n). An adjacent block is a block of the form (i, i+1, i+2, ...).

%C Sum of entries in row n = A000110(n) (the Bell numbers).

%H Alois P. Heinz, <a href="/A177254/b177254.txt">Rows n = 0..140, flattened</a>

%F The row generating polynomial P[n](t) is given by P[n](t)=Q[n](1,t,t), where Q[n](u,v,w) is obtained recursively from Q[n](u,v,w) =u(dQ[n-1]/du)_{w=v} + u(dQ[n-1]/dv)_{w=v} + w(dQ[n-1]/dw) + w(Q[n-1])_{w=v}, Q[0]=1. Here Q[n](u,v,w) is the trivariate generating polynomial of the partitions of {1,2,...,n}, where u marks blocks that are not adjacent, v marks adjacent blocks not ending with n, and w marks adjacent blocks ending with n.

%F T(n, 0) = A168444(n).

%F Sum_{k=0..n} T(n, k) = A000110(n) (row sums).

%F Sum_{k=0..n} k*T(n, k) = A177255(n).

%F From _G. C. Greubel_, May 12 2024: (Start)

%F T(n, n) = 1.

%F T(n, n-1) = n-1, for n >= 1.

%F T(n, n-2) = A002378(n-2), for n >= 2.

%F T(n, n-3) = A162148(n-3), for n >= 3.

%F T(n, n-4) = A302560(n-3), for n >= 4. (End)

%e T(4,2)=6 because we have 1-234, 12-34, 123-4, 13-2-4, 14-2-3, and 1-24-3.

%e Triangle starts:

%e      1;

%e      0,    1;

%e      0,    1,    1;

%e      0,    2,    2,    1;

%e      1,    4,    6,    3,    1;

%e      5,   13,   17,   12,    4,   1;

%e     21,   51,   61,   44,   20,   5,   1;

%e     91,  219,  255,  185,   90,  30,   6,  1;

%e    422, 1019, 1182,  867,  440, 160,  42,  7,  1;

%e   2103, 5108, 5964, 4430, 2322, 896, 259, 56,  8,  1;

%p Q[0] := 1: for n to 10 do Q[n] := expand(u*subs(w = v, diff(Q[n-1], u))+u*subs(w = v, diff(Q[n-1], v))+w*(diff(Q[n-1], w))+w*subs(w = v, Q[n-1])) end do: for n from 0 to 10 do P[n] := sort(expand(subs({v = t, w = t, u = 1}, Q[n]))) end do; for n from 0 to 10 do seq(coeff(P[n], t, j), j = 0 .. n) end do; # yields sequence in triangular form

%Y Cf. A000110, A002378, A162148, A168444, A177255, A177256, A177257.

%Y Cf. A302560.

%K nonn,tabl

%O 0,8

%A _Emeric Deutsch_, May 07 2010