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A213280 Triangle read by rows: T(n,k) (n>=1, 1 <= k <= n) = number of permutations of [1..n] in which none of the cycle lengths are divisible by k. 2

%I

%S 0,0,1,0,3,4,0,9,16,18,0,45,80,90,96,0,225,400,540,576,600,0,1575,

%T 2800,3780,4032,4200,4320,0,11025,22400,26460,32256,33600,34560,35280,

%U 0,99225,179200,238140,290304,302400,311040,317520,322560,0,893025,1792000,2381400,2612736,3024000,3110400,3175200,3225600,3265920

%N Triangle read by rows: T(n,k) (n>=1, 1 <= k <= n) = number of permutations of [1..n] in which none of the cycle lengths are divisible by k.

%H Alois P. Heinz, <a href="/A213280/b213280.txt">Rows n = 1..141, flattened</a>

%H E. D. Bolker and A. M. Gleason, <a href="http://dx.doi.org/10.1016/0097-3165(80)90012-6">Counting permutations</a>, J. Combin. Thy., A 29 (1980), 236-242.

%e Triangle begins

%e [0],

%e [0, 1],

%e [0, 3, 4],

%e [0, 9, 16, 18],

%e [0, 45, 80, 90, 96],

%e [0, 225, 400, 540, 576, 600],

%e [0, 1575, 2800, 3780, 4032, 4200, 4320],

%e [0, 11025, 22400, 26460, 32256, 33600, 34560, 35280],

%e [0, 99225, 179200, 238140, 290304, 302400, 311040, 317520, 322560],

%e [0, 893025, 1792000, 2381400, 2612736, 3024000, 3110400, 3175200, 3225600, 3265920],

%e [0, 9823275, 19712000, 26195400, 28740096, 33264000, 34214400, 34927200, 35481600, 35925120, 36288000],

%e [0, 108056025, 216832000, 288149400, 344881152, 365904000, 410572800, 419126400, 425779200, 431101440, 435456000, 439084800],

%e ...

%p read transforms;

%p f:=(n,d)->mul(j-did(j,d),j=1..n); # did(d,j) = 1 iff j divides d, otherwise 0

%p g:=n->[seq(f(n,d),d=1..n)];

%p [seq(g(n),n=1..14)];

%p # second Maple program:

%p T:= proc(n, k) option remember; `if`(n=0, 1, add(

%p `if`(irem(j, k)=0, 0, binomial(n-1, j-1)*(j-1)!*

%p T(n-j, k)), j=1..n))

%p end:

%p seq(seq(T(n, k), k=1..n), n=1..12); # _Alois P. Heinz_, May 14 2016

%t T[n_, k_] := T[n, k] = If[n == 0, 1, Sum[If[Mod[j, k] == 0, 0, Binomial[n - 1, j - 1]*(j - 1)!*T[n - j, k]], {j, 1, n}]];

%t Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-Fran├žois Alcover_, May 26 2016, after _Alois P. Heinz_ *)

%Y Cf. A001563 (diagonal of triangle), A213279.

%K nonn,tabl

%O 1,5

%A _N. J. A. Sloane_, Jun 08 2012

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Last modified April 4 23:59 EDT 2020. Contains 333238 sequences. (Running on oeis4.)