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A080510 Triangle read by rows: T(n,k) gives the number of set partitions of {1,...,n} with maximum block length k. 28
1, 1, 1, 1, 3, 1, 1, 9, 4, 1, 1, 25, 20, 5, 1, 1, 75, 90, 30, 6, 1, 1, 231, 420, 175, 42, 7, 1, 1, 763, 2016, 1015, 280, 56, 8, 1, 1, 2619, 10024, 6111, 1890, 420, 72, 9, 1, 1, 9495, 51640, 38010, 12978, 3150, 600, 90, 10, 1, 1, 35695, 276980, 244035, 91938, 24024, 4950, 825, 110, 11, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

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

From Peter Luschny, 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)

From Gary W. Adamson, Feb 24 2011: (Start)

Construct an array in which n-th row is the partition function G(n,k), where G(n,1),...G(n,6) = A000012, A000085, A001680, A001681, A110038, A148092, with the first few rows

1,...1,...1,...1,...1,...1,....1,... = A000012

1,...2,...4,..10,..26,..76,..232,... = A000085

1,...2,...5,..14,..46,.166,..652,... = A001680

1,...2,...5,..15,..51,.196,..827,... = A001681

1,...2....5...15...52..202...869,... = A110038

1,...2,...5...15...52..203...876.... = A148092

...

Rows tend to A000110, the Bell numbers. Taking finite differences from the top, then reorienting, we obtain triangle A080510.

The n-th row of the array is the eigensequence of an infinite lower triangular matrix with n diagonals of Pascal's triangle starting from the right and the rest zeros. (End)

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

Peter Luschny, Counting with Partitions. - Peter Luschny, Mar 09 2009

Peter Luschny, Generalized Stirling_2 Triangles. - Peter Luschny, Mar 09 2009

FORMULA

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

From Peter Luschny, 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)

EXAMPLE

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}}.

Triangle begins:

1;

1,    1;

1,    3,     1;

1,    9,     4,    1;

1,   25,    20,    5,    1;

1,   75,    90,   30,    6,   1;

1,  231,   420,  175,   42,   7,  1;

1,  763,  2016, 1015,  280,  56,  8,  1;

1, 2619, 10024, 6111, 1890, 420, 72,  9,  1;

MAPLE

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

       add(b(n-i*j, i-1) *n!/i!^j/(n-i*j)!/j!, j=0..n/i)))

    end:

T:= (n, k)-> b(n, k) -b(n, k-1):

seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Apr 20 2012

MATHEMATICA

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

(* Second program: *)

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n-i*j, i-1]*n!/i!^j/(n-i*j)!/j!, {j, 0, n/i}]]]; T[n_, k_] := b[n, k]-b[n, k-1]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-Fran├žois Alcover, Feb 25 2014, after Alois P. Heinz *)

CROSSREFS

Cf. A080107, A080337, A008277, A276922.

Cf. A157396, A157397, A157398, A157399, A157400, A157401, A157402, A157403, A157404, A157405. - Peter Luschny, Mar 09 2009

Cf. A000012, A000085, A001680, A001681, A110038, A148092. - Gary W. Adamson, Feb 24 2011

Columns k=1..10 give: A000012 (for n>0), A001189, A229245, A229246, A229247, A229248, A229249, A229250, A229251, A229252. - Alois P. Heinz, Sep 17 2013

T(2n,n) gives A276961.

Sequence in context: A152570 A100537 A069605 * A124496 A074881 A142992

Adjacent sequences:  A080507 A080508 A080509 * A080511 A080512 A080513

KEYWORD

nonn,tabl

AUTHOR

Wouter Meeussen, Mar 22 2003

STATUS

approved

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Last modified April 28 12:25 EDT 2017. Contains 285575 sequences.