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A186754
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Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k increasing cycles (0<=k<=n). A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1) < b(2) < b(3) < ... .
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8
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1, 0, 1, 0, 1, 1, 1, 1, 3, 1, 5, 5, 7, 6, 1, 23, 36, 25, 25, 10, 1, 129, 234, 166, 110, 65, 15, 1, 894, 1597, 1316, 686, 385, 140, 21, 1, 7202, 12459, 10893, 5754, 2611, 1106, 266, 28, 1, 65085, 111451, 97287, 54559, 22428, 8841, 2730, 462, 36, 1, 651263, 1116277, 963121, 554670, 229405, 80871, 26397, 6000, 750, 45, 1
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OFFSET
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0,9
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COMMENTS
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Sum of entries in row n is n!.
Sum_{k=0..n} k*T(n,k) = A002627(n).
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LINKS
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FORMULA
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E.g.f.: G(t,z) = exp((t-1)(exp(z)-1))/(1-z).
The 4-variate e.g.f. H(u,v,w,z) of permutations with respect to size (marked by z), number of fixed points (marked by u), number of increasing cycles of length >=2 (marked by v), and number of nonincreasing cycles (marked by w) is given by H(u,v,w,z) = exp(uz+v(exp(z)-1-z)+w(1-exp(z))/(1-z)^w. Remark: the nonincreasing cycles are necessarily of length >=3. We have: G(t,z)=H(t,t,1,z).
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EXAMPLE
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T(3,0)=1 because we have (132).
T(4,2)=7 because we have (1)(234), (13)(24), (12)(34), (123)(4), (124)(3), (134)(2), and (14)(23).
Triangle starts:
1;
0, 1;
0, 1, 1;
1, 1, 3, 1;
5, 5, 7, 6, 1;
23, 36, 25, 25, 10, 1;
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MAPLE
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G := exp((t-1)*(exp(z)-1))/(1-z); Gser := simplify(series(G, z = 0, 16)): for n from 0 to 10 do P[n] := sort(expand(factorial(n)*coeff(Gser, z, n))) end do: for n from 0 to 10 do seq(coeff(P[n], t, j), j = 0 .. n) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(
b(n-i)*binomial(n-1, i-1)*(x+(i-1)!-1), i=1..n)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
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MATHEMATICA
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b[n_] := b[n] = Expand[If[n==0, 1, Sum[b[n-i]*Binomial[n-1, i-1]*(x + (i-1)! - 1), {i, 1, n}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Oct 04 2016, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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