A108458 - OEIS

%I #25 Dec 10 2024 10:00:51

%S 1,0,1,0,1,2,0,1,3,5,0,1,5,10,15,0,1,9,22,37,52,0,1,17,52,99,151,203,

%T 0,1,33,130,283,471,674,877,0,1,65,340,855,1561,2386,3263,4140,0,1,

%U 129,922,2707,5451,8930,12867,17007,21147,0,1,257,2572,8919,19921,35098,53411,73681,94828,115975

%N Triangle read by rows: T(n,k) is the number of set partitions of {1,2,...,n} in which the last block is the singleton {k}, 1<=k<=n; the blocks are ordered with increasing least elements.

%C Another way to obtain this sequence (with offset 0): Form the infinite array U(n,k) = number of labeled partitions of (n,k) into pairs (i,j), for n >= 0, k >= 0 and read it by antidiagonals. In other words, U(n,k) = number of partitions of n black objects labeled 1..n and k white objects labeled 1..k. Each block must have at least one white object.

%C Then T(n,k)=U(n+k,k+1). Thus the two versions are related like "multichoose" to "choose". - _Augustine O. Munagi_, Jul 16 2007

%H Alois P. Heinz, <a href="/A108458/b108458.txt">Rows n = 1..141, flattened</a>

%F T(n,1)=0 for n>=2; T(n,2)=1 for n>=2; T(n,3)=1+2^(n-3) for n>=3; T(n,n)=B(n-1), T(n,n-1)=B(n-1)-B(n-2), where B(q) are the Bell numbers (A000110).

%F Double e.g.f.: exp(exp(x)*(exp(y)-1)).

%F U(n,k) = Sum_{i=0..k} i^(n-k)*Stirling2(k,i). - _Vladeta Jovovic_, Jul 12 2007

%e Triangle T(n,k) starts:

%e   1;

%e   0,1;

%e   0,1,2;

%e   0,1,3,5;

%e   0,1,5,10,15;

%e T(5,3)=5 because we have 1245|3, 145|2|3, 14|25|3, 15|24|3 and 1|245|3.

%e The arrays U(n,k) starts:

%e    1  0  0   0   0 ...

%e    1  1  1   1   1 ...

%e    2  3  5   9  17 ...

%e    5 10 22  52 130 ...

%e   15 37 99 283 855 ...

%t T[n_, k_] := Sum[If[n == k, 1, i^(n-k)]*StirlingS2[k, i], {i, 0, k}];

%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 10 2024, after _Vladeta Jovovic_ *)

%Y Row sums of T(n, k) yield A124496(n, 1).

%Y Cf. A108461.

%Y Columns of U(n, k): A000110, A005493, A033452.

%Y Rows of U(n, k): A000007, A000012, A000051.

%Y Main diagonal: A108459.

%K nonn,tabl

%O 1,6

%A _Christian G. Bower_, Jun 03 2005; _Emeric Deutsch_, Nov 14 2006

%E Edited by _N. J. A. Sloane_, May 22 2008, at the suggestion of _Vladeta Jovovic_. This entry is a composite of two entries submitted independently by _Christian G. Bower_ and _Emeric Deutsch_, with additional comments from _Augustine O. Munagi_.