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A185982 Triangle read by rows: number of set partitions of n elements with k connectors, 0<=k

%I A185982 #45 May 09 2023 16:31:13
%S A185982 1,1,1,1,3,1,1,7,6,1,1,16,24,10,1,1,39,86,61,15,1,1,105,307,313,129,
%T A185982 21,1,1,314,1143,1520,891,242,28,1,1,1035,4513,7373,5611,2161,416,36,
%U A185982 1,1,3723,18956,36627,34213,17081,4658,670,45,1,1,14494,84546,188396,208230,127540,45095,9187,1025,55,1
%N A185982 Triangle read by rows: number of set partitions of n elements with k connectors, 0<=k<n.
%H A185982 Alois P. Heinz, <a href="/A185982/b185982.txt">Rows n = 1..141, flattened</a>
%H A185982 T. Mansour and A. O. Munagi, <a href="http://dx.doi.org/10.1016/j.ejc.2009.07.001">Block-connected set partitions</a>, European J. Combin., 31 (2010), 887-902.
%e A185982 A connector is a pair (a, a+1) in a set partition if a is in block i and a+1 is in block i+1, for some i.  For example a(4,1) = 7, counting 1/234, 13/2/4, 14/23, 134/2, 12/34, 124/3, 123/4.
%e A185982 Triangle begins:
%e A185982   1;
%e A185982   1,   1;
%e A185982   1,   3,   1;
%e A185982   1,   7,   6,   1;
%e A185982   1,  16,  24,  10,   1;
%e A185982   1,  39,  86,  61,  15,  1;
%e A185982   1, 105, 307, 313, 129, 21, 1;
%e A185982   ...
%p A185982 b:= proc(n, i, m) option remember; `if`(n=0, 1, add(expand(
%p A185982        b(n-1, j, max(m, j))*`if`(j=i+1, x, 1)), j=1..m+1))
%p A185982     end:
%p A185982 T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(b(n, 1, 0)):
%p A185982 seq(T(n), n=1..12);  # _Alois P. Heinz_, Mar 25 2016
%t A185982 b[n_, i_, m_] := b[n, i, m] = If[n == 0, 1, Sum[b[n-1, j, Max[m, j]]*If[j == i+1, x, 1], {j, 1, m+1}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n-1}]][b[n, 1, 0]]; Table[T[n], {n, 1, 12}] // Flatten (* _Jean-François Alcover_, Apr 13 2016, after _Alois P. Heinz_ *)
%Y A185982 Columns k=0-10 give: A000012, A271788, A271789, A271790, A271791, A271792, A271793, A271794, A271795, A271796, A271797.
%Y A185982 Row sums give A000110.
%Y A185982 T(n+1,n-1) gives A000217.
%Y A185982 T(2n,n) gives A271841.
%Y A185982 Cf. A185983, A270953, A271206, A271270, A271271, A272064.
%K A185982 nonn,tabl
%O A185982 1,5
%A A185982 _Brian Drake_, Feb 08 2011
%E A185982 More terms from _Alois P. Heinz_, Oct 11 2011