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A080510
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Triangle read by rows: T(n,k) gives the number of set partitions of {1,..,n} with maximum block length k.
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16
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| 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)
Contribution from Gary W. Adamson, (qntmpkt(AT)yahoo.com), 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)
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LINKS
| 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]
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FORMULA
| 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)
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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}}. Sequence starts as 1; 1,1; 1,3,1; 1,9,4,1;
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MATHEMATICA
| << DiscreteMath`NewCombinatorica`; Table[Length/@Split[Sort[Max[Length/@# ]&/@SetPartitions[n]]], {n, 12}]
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CROSSREFS
| 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]
Cf. A000012, A000085, A001680, A001681, A110038, A148092 [from Gary W. Adamson, (qntmpkt(AT)yahoo.com, Feb 24, 2011]
Sequence in context: A152570 A100537 A069605 * A124496 A074881 A142992
Adjacent sequences: A080507 A080508 A080509 * A080511 A080512 A080513
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KEYWORD
| nonn,tabl
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AUTHOR
| Wouter Meeussen (wouter.meeussen(AT)pandora.be), Mar 22 2003
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