A224211 Irregular triangular array read by rows. T(n,k) is the number of n-permutations with exactly k distinct cycle lengths; n>=1, 1<=k<=floor( (-1+(1+8n)^(1/2))/2 ).
1, 1, 2, 3, 6, 8, 24, 50, 120, 234, 120, 720, 1764, 630, 5040, 11808, 7392, 40320, 109584, 69552, 362880, 954000, 763200, 151200, 3628800, 10628640, 8165520, 1330560, 39916800, 113891040, 109010880, 25280640, 479001600, 1486442880, 1345687200, 381775680, 6227020800, 18913184640, 19773804960, 6763236480, 87178291200, 283465647360, 291950568000, 102508005600, 10897286400
Offset: 1
Examples
: 1; : 1; : 2, 3; : 6, 8; : 24, 50; : 120, 234, 120; : 720, 1764, 630; : 5040, 11808, 7392; : 40320, 109584, 69552; : 362880, 954000, 763200, 151200;
Links
- Alois P. Heinz, Rows n = 1..300, flattened
Programs
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Maple
b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+ (i-1)!*b(n-i, i-1)* `if`(i>n, 0, binomial(n, i)*x)))) end: T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)): seq(T(n), n=1..15); # Alois P. Heinz, Oct 21 2015 -
Mathematica
nn=15;f[list_]:=Select[list,#>0&];Map[f,Drop[Range[0,nn]!CoefficientList[Series[Product[(1+y x^i/i),{i,1,nn}],{x,0,nn}],{x,y}],1]]//Grid
Formula
E.g.f.: Product_{i>=1} (1 + y*x)^i/i.
Comments