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A180013 Triangular array read by rows: T(n,k) = number of fixed points in the permutations of {1,2,...,n} that have exactly k cycles; n>=1, 1<=k<=n. 1

%I

%S 1,0,2,0,3,3,0,8,12,4,0,30,55,30,5,0,144,300,210,60,6,0,840,1918,1575,

%T 595,105,7,0,5760,14112,12992,5880,1400,168,8,0,45360,117612,118188,

%U 60921,17640,2898,252,9,0,403200,1095840,1181240,672840,224490,45360,5460,360,10

%N Triangular array read by rows: T(n,k) = number of fixed points in the permutations of {1,2,...,n} that have exactly k cycles; n>=1, 1<=k<=n.

%C Row sums = n! which is the number of fixed points in all the permutations of {1,2,...,n}.

%C It appears that column k = 2 is A001048 (with different offset).

%C From _Olivier Gérard_, Oct 23 2012: (Start)

%C This is a multiple of the triangle of Stirling numbers of the first kind, A180013(n,k) = (n)*A132393(n-1,k).

%C Another interpretation is : T(n,n-k) is the total number of ways to insert the symbol n among the cycles of permutations of [n-1] with (n+1-k) cycles to form a canonical cycle representation of a permutation of [n]. For each cycle of length c, there are c places to insert a symbol, and for each permutation there is the possibility to create a new cycle (a fixed point).

%C (End)

%H G. C. Greubel, <a href="/A180013/b180013.txt">Table of n, a(n) for n = 1..5050</a>

%F E.g.f.: for column k: x*(log(1/(1-x)))^(k-1)/(k-1)!.

%F T(n, k) = [x^k] (n+1)!*hypergeom([-n,1-x],[1],1)) for n>0. # _Peter Luschny_, Jan 28 2016

%e T(4,3)= 12 because there are 12 fixed points in the permutations of 4 that have 3 cycles: (1)(2)(4,3); (1)(3,2)(4); (1)(4,2)(3); (2,1)(3)(4); (3,1)(2)(4); (4,1)(2)(3) where the permutations are represented in their cycle notation.

%e 1

%e 0 2

%e 0 3 3

%e 0 8 12 4

%e 0 30 55 30 5

%e 0 144 300 210 60 6

%e 0 840 1918 1575 595 105 7

%p egf:= k-> x * (log(1/(1-x)))^(k-1) / (k-1)!:

%p T:= (n,k)-> n! * coeff(series(egf(k), x, n+1), x, n):

%p seq(seq(T(n, k), k=1..n), n=1..10); # _Alois P. Heinz_, Jan 16 2011

%p # As coefficients of polynomials:

%p with(PolynomialTools): with(ListTools): A180013_row := proc(n)

%p `if`(n=0, 1,(n+1)!*hypergeom([-n,1-x],[1],1)); CoefficientList(simplify(%),x) end: FlattenOnce([seq(A180013_row(n), n=0..9)]); # _Peter Luschny_, Jan 28 2016

%t Flatten[Table[Table[(n + 1) Abs[StirlingS1[n, k]], {k, 0, n}], {n, 0, 9}],1] (* _Olivier Gérard_, Oct 23 2012 *)

%Y Cf. A000142, A001048. Diagonal, lower diagonal give: A000027, A027480(n+1).

%K nonn,tabl

%O 1,3

%A _Geoffrey Critzer_, Jan 13 2011

%E More terms from _Alois P. Heinz_, Jan 16 2011

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Last modified February 19 19:06 EST 2020. Contains 332047 sequences. (Running on oeis4.)