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A185983 Triangle read by rows: number of set partitions of n elements with k circular connectors. 5
1, 1, 0, 1, 0, 1, 1, 0, 3, 1, 1, 0, 8, 4, 2, 1, 1, 20, 15, 14, 1, 1, 6, 53, 61, 68, 11, 3, 1, 25, 159, 267, 295, 97, 32, 1, 1, 93, 556, 1184, 1339, 694, 242, 28, 3, 1, 346, 2195, 5366, 6620, 4436, 1762, 371, 48, 2, 1, 1356, 9413, 25400, 34991, 27497, 12977, 3650, 634, 53, 3 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,9
COMMENTS
A pair (a,a+1) in a set partition with m blocks is a circular connector if a is in block i and a+1 is in block (i mod m)+1 for some i. In addition, (n,1) is considered a circular connector if n is in block m.
LINKS
T. Mansour and A. O. Munagi, Block-connected set partitions, European J. Combin., 31 (2010), 887-902.
EXAMPLE
For a(4,2) = 8, the set partitions are 1/234, 134/2, 124/3, 123/4, 12/34, 14/23, 1/24/3, and 13/2/4.
For a(5,1) = 1, the set partition is 13/25/4.
For a(6,6) = 3, the set partitions are 135/246, 14/25/36, 1/2/3/4/5/6.
Triangle begins:
1;
1, 0;
1, 0, 1;
1, 0, 3, 1;
1, 0, 8, 4, 2;
1, 1, 20, 15, 14, 1;
1, 6, 53, 61, 68, 11, 3;
...
MAPLE
b:= proc(n, i, m, t) option remember; `if`(n=0, x^(t+
`if`(i=m and m<>1, 1, 0)), add(expand(b(n-1, j,
max(m, j), `if`(j=m+1, 0, t+`if`(j=1 and i=m
and j<>m, 1, 0)))*`if`(j=i+1, x, 1)), j=1..m+1))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 1, 0$2)):
seq(T(n), n=0..12); # Alois P. Heinz, Mar 30 2016
MATHEMATICA
b[n_, i_, m_, t_] := b[n, i, m, t] = If[n == 0, x^(t + If[i == m && m != 1, 1, 0]), Sum[Expand[b[n - 1, j, Max[m, j], If[j == m + 1, 0, t + If[j == 1 && i == m && j != m, 1, 0]]]*If[j == i + 1, x, 1]], {j, 1, m + 1}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 1, 0, 0] ];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 19 2016, after Alois P. Heinz *)
CROSSREFS
Cf. A185982. Row sums are A000110.
T(n,n) = A032741(n) if n>0. - Alois P. Heinz, Oct 14 2011
T(2n,n) gives A362944.
Sequence in context: A143295 A330892 A289978 * A179742 A285000 A321444
KEYWORD
nonn,tabl
AUTHOR
Brian Drake, Feb 08 2011
EXTENSIONS
More terms from Alois P. Heinz, Oct 14 2011
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)