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A126074
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Triangle read by rows: T(n,k) is the number of permutations of n elements that have the longest cycle length k.
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6
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1, 1, 1, 1, 3, 2, 1, 9, 8, 6, 1, 25, 40, 30, 24, 1, 75, 200, 180, 144, 120, 1, 231, 980, 1260, 1008, 840, 720, 1, 763, 5152, 8820, 8064, 6720, 5760, 5040
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| Sum of the n-th row is the number of all permutations of n elements: Sum_{k=1..n, T(n,k)} = n! = A000142(n) We can extend T(n,k)=0, if k<=0 or k>n.
Contribution from Peter Luschny (peter(AT)luschny.de), Mar 07 2009: (Start)
Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = -1,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A102189.
Same partition product with length statistic is A008275.
Diagonal a(A000217(n)) = rising_factorial(1,n-1), A000142(n-1) (n > 0).
Row sum is A000142. (End)
Let k in {1,2,3,...} index the family of sequences A000012,A000085,A057693, A070945,A070946,A070947,... respectively. Column k is the k-th sequence minus its immediate predecessor. For example, T(5,3)=A057693(5)-A000085(5) [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), May 23 2009]
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LINKS
| IBM Research : Ponder This
Peter Luschny, Counting with Partitions. [From Peter Luschny (peter(AT)luschny.de), Mar 07 2009]
Peter Luschny, Generalized Stirling_1 Triangles. [From Peter Luschny (peter(AT)luschny.de), Mar 07 2009]
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FORMULA
| T(n,1) = 1 T(n,2) = n! * Sum_{k=1..[n/2], (1/(k! * (2!)^k * (n-2k)!)} T(n,k) = n!/k * (1-1/(n-k)-...-1/(k+1)-1/2k), if n/3 < k <= n/2 T(n,k) = n!/k, if n/2 < k <= n T(n,n) = (n-1)! = A000142(n-1)
E.g.f. for k-th column: exp(-x^k*LerchPhi(x,1,k))*(exp(x^k/k)-1)/(1-x). - Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 03 2007
Contribution from Peter Luschny (peter(AT)luschny.de), Mar 07 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-2}(j-n+1). (End)
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EXAMPLE
| 1,
1, 1,
1, 3, 2,
1, 9, 8, 6,
1, 25, 40, 30, 24,
1, 75, 200, 180, 144, 120,
1, 231, 980, 1260, 1008, 840, 720,
1, 763, 5152, 8820, 8064, 6720, 5760, 5040,
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MATHEMATICA
| Table[CoefficientList[ Series[(Exp[x^m/m] - 1) Exp[Sum[x^k/k, {k, 1, m - 1}]], {x, 0, 8}], x]*Table[n!, {n, 0, 8}], {m, 1, 8}] // Transpose // Grid [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), May 23 2009]
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CROSSREFS
| Cf. A000142.
Cf. A071007, A080510.
Cf. A157386, A157385, A157384, A157383, A157400, A157391, A157392, A157393, A157394, A157395 [From Peter Luschny (peter(AT)luschny.de), Mar 07 2009]
Sequence in context: A155788 A108073 A057731 * A108916 A119421 A121581
Adjacent sequences: A126071 A126072 A126073 * A126075 A126076 A126077
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KEYWORD
| base,nonn,tabl
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AUTHOR
| Dan Dima (dimad72(AT)gmail.com), Mar 01 2007
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