A218868
Triangular array read by rows: T(n,k) is the number of n-permutations that have exactly k distinct cycle lengths.
Original entry on oeis.org
1, 2, 3, 3, 10, 14, 25, 95, 176, 424, 120, 721, 3269, 1050, 6406, 21202, 12712, 42561, 178443, 141876, 436402, 1622798, 1418400, 151200, 3628801, 17064179, 17061660, 2162160, 48073796, 177093256, 212254548, 41580000, 479001601, 2293658861, 2735287698, 719072640
Offset: 1
: 1;
: 2;
: 3, 3;
: 10, 14;
: 25, 95;
: 176, 424, 120;
: 721, 3269, 1050;
: 6406, 21202, 12712;
: 42561, 178443, 141876;
: 436402, 1622798, 1418400, 151200;
-
with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
`if`(i<1, 0, add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
b(n-i*j, i-1)*`if`(j=0, 1, x), j=0..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):
seq(T(n), n=1..16); # Alois P. Heinz, Aug 21 2014
-
nn=10;a=Product[1-y+y Exp[x^i/i],{i,1,nn}];f[list_]:=Select[list,#>0&];Map[f,Drop[Range[0,nn]!CoefficientList[Series[a ,{x,0,nn}],{x,y}],1]]//Grid
A088142
Number of partitions of n-set with 2 block sizes.
Original entry on oeis.org
3, 10, 50, 116, 560, 1730, 6123, 30122, 116908, 507277, 2492737, 15328119, 56182092, 441156796, 2093130576, 15965840718, 77353276330, 693400983344, 3517825829117, 35126205660152, 187347585491624, 1952969742765476
Offset: 3
-
with(numtheory): with(combinat):
a:= n-> add(add(add(multinomial(n, i$j, d$((n-i*j)/d))/j!/((n-i*j)/d)!,
d=select(x->xAlois P. Heinz, Feb 01 2014
-
max = 25; G[x_] = Sum[Exp[x^k/k!]-1, {k, 1, max}]; H[x_] = Sum[(Exp[x^k/k!]-1)^2, {k, 1, max}]; Drop[CoefficientList[(G[x]^2-H[x])/2 + O[x]^max, x]*Range[0, max-1]!, 3] (* Jean-François Alcover, Jul 01 2015 *)
A131661
Number of compositions of n such that the cardinality of the set of parts is 2.
Original entry on oeis.org
0, 0, 2, 5, 14, 22, 44, 68, 107, 172, 261, 396, 606, 950, 1414, 2238, 3418, 5411, 8368, 13297, 20840, 33268, 52549, 84120, 133775, 214611, 343025, 551064, 883600, 1421767, 2284870, 3680296, 5924725, 9551161, 15393855, 24834827, 40061700
Offset: 1
Cf.
A242900 (with distinct multiplicities).
-
with(numtheory):
a:= n-> add(add(add(binomial(j+(n-i*j)/d, j), d=select(x->xAlois P. Heinz, Feb 01 2014
-
Rest@ CoefficientList[ Series[ Sum[ x^(i + j)*(x^i + x^j - 2)/((x^i - 1)*(x^j - 1)*(x^i + x^j - 1)), {i, 2, 37}, {j, i - 1}], {x, 0, 37}], x] (* Robert G. Wilson v, Sep 16 2007 *)
A133118
Number of partitions of n-set with 3 block sizes.
Original entry on oeis.org
60, 315, 2268, 14742, 72180, 464640, 2676366, 16400098, 94209206, 673282610, 4095231104, 29371828846, 197547348216, 1513916607683, 10904464442572, 87070803499372, 673555061736062, 5718121102062336, 47028289679340734, 418812093667530755, 3680961843042545490, 34161428275433710485
Offset: 6
-
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[multinomial[n, Prepend[Table[i, {j}], n - i*j]]/j!*b[n - i*j, i - 1]*If[j == 0, 1, x], {j, 0, n/i}]]];
a[n_] := Coefficient[b[n, n], x, 3];
Array[a, 22, 6] (* Jean-François Alcover, May 24 2019, after Alois P. Heinz in A208437 *)
A133119
Number of permutations of [n] with 3 cycle lengths.
Original entry on oeis.org
120, 1050, 12712, 141876, 1418400, 17061660, 212254548, 2735287698, 37354035628, 581350330470, 8895742806480, 151305163230480, 2659183039338192, 50112909523522476, 976443721325014300, 20413628375979803370, 434137453618439716068
Offset: 6
Showing 1-5 of 5 results.
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