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A186761
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Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k increasing odd 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)<... . A cycle is said to be odd if it has an odd number of entries. For example, the permutation (152)(347)(6)(8) has 3 increasing odd cycles.
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10
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1, 0, 1, 1, 0, 1, 1, 4, 0, 1, 9, 4, 10, 0, 1, 33, 56, 10, 20, 0, 1, 235, 218, 211, 20, 35, 0, 1, 1517, 1982, 833, 616, 35, 56, 0, 1, 12593, 14040, 9612, 2408, 1526, 56, 84, 0, 1, 111465, 134248, 72588, 35176, 5838, 3360, 84, 120, 0, 1, 1122819, 1305126, 797461, 276120, 107710, 12516, 6762, 120, 165, 0, 1
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OFFSET
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0,8
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COMMENTS
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Sum of entries in row n is n!.
T(n,0)=A186762(n).
Sum(k*T(n,k),k=0..n) = A186763(n).
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LINKS
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Table of n, a(n) for n=0..65.
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FORMULA
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E.g.f.: G(t,z)=exp((t-1)sinh z)/(1-z).
The 5-variate e.g.f. H(x,y,u,v,z) of permutations with respect to size (marked by z), number of increasing odd cycles (marked by x), number of increasing even cycles (marked by y), number of nonincreasing odd cycles (marked by u), and number of nonincreasing even cycles (marked by v), is given by
H(x,y,u,v,z)=exp(((x-u)sinh z + (y-v)(cosh z - 1))*(1+z)^{(u-v)/2}/(1-z)^{(u+v)/2}.
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EXAMPLE
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T(3,1)=4 because we have (1)(23), (12)(3), (13)(2), and (123).
T(4,1)=4 because we have (1)(243), (143)(2), (142)(3), and (132)(4).
Triangle starts:
1;
0,1;
1,0,1;
1,4,0,1;
9,4,10,0,1;
33,56,10,20,0,1;
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MAPLE
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g := exp((t-1)*sinh(z))/(1-z): gser := simplify(series(g, z = 0, 13)): 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, k), k = 0 .. n) end do; # yields sequence in triangular form
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CROSSREFS
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Cf. A186762, A186763, A186764, A186766, A186769
Sequence in context: A079642 A221483 A121408 * A199786 A189245 A121301
Adjacent sequences: A186758 A186759 A186760 * A186762 A186763 A186764
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch, Feb 27 2011
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STATUS
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approved
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