1,5

Row sums are A000110 (Bell numbers). Second column is A001189 (Degree n permutations of order exactly 2).

Contribution from Peter Luschny (peter(AT)luschny.de), Mar 09 2009: (Start)

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = -1,

summed over parts with equal biggest part (see the Luschny link).

Underlying partition triangle is A036040.

Same partition product with length statistic is A008277.

Diagonal a(A000217) = A000012.

Row sum is A000110. (End)

Peter Luschny, Counting with Partitions. [From Peter Luschny (peter(AT)luschny.de), Mar 09 2009]

Peter Luschny, Generalized Stirling_2 Triangles. [From Peter Luschny (peter(AT)luschny.de), Mar 09 2009]

E.g.f. for k-th column: exp(exp(x)*GAMMA(k, x)/(k-1)!-1)*(exp(x^k/k!)-1). - Vladeta Jovovic (vladeta(AT)eunet.rs), Feb 04 2005

Contribution from Peter Luschny (peter(AT)luschny.de), Mar 09 2009: (Start)

T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n

T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that

1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),

f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-1}(-1) = (-1)^n. (End)

T[4,3]= 4 since there are 4 set partitions with longest block of length 3 : {{1},{2,3,4}}, {{1,3,4},{2}}, {{1,2,3},{4}} and {{1,2,4},{3}}. Sequence starts as 1; 1,1; 1,3,1; 1,9,4,1;

<< DiscreteMath`NewCombinatorica`; Table[Length/@Split[Sort[Max[Length/@# ]&/@SetPartitions[n]]], {n, 12}]

Cf. A080107, A080337, A008277.

Cf. A157396, A157397, A157398, A157399, A157400, A157401, A157402, A157403, A157404, A157405 [From Peter Luschny (peter(AT)luschny.de), Mar 09 2009]

Sequence in context: A152570 A100537 A069605 this_sequence A124496 A074881 A142992

Adjacent sequences: A080507 A080508 A080509 this_sequence A080511 A080512 A080513

nonn,tabl

Wouter Meeussen (wouter.meeussen(AT)pandora.be), Mar 22 2003