cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A074761 Number of partitions of n of order n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 9, 1, 4, 5, 1, 1, 12, 1, 27, 7, 6, 1, 81, 1, 7, 1, 54, 1, 407, 1, 1, 11, 9, 13, 494, 1, 10, 13, 423, 1, 981, 1, 137, 115, 12, 1, 1309, 1, 59, 17, 193, 1, 240, 21, 1207, 19, 15, 1, 47274, 1, 16, 239, 1, 25, 3284, 1, 333, 23, 3731, 1, 42109, 1, 19
Offset: 1

Views

Author

Vladeta Jovovic, Sep 28 2002

Keywords

Comments

Order of partition is lcm of its parts.
a(n) is the number of conjugacy classes of the symmetric group S_n such that a representative of the class has order n. Here order means the order of an element of a group. Note that a(n) = 1 if and only if n is a prime power. - W. Edwin Clark, Aug 05 2014

Examples

			The a(15) = 5 partitions are (15), (5,3,3,3,1), (5,5,3,1,1), (5,3,3,1,1,1,1), (5,3,1,1,1,1,1,1,1). - _Gus Wiseman_, Aug 01 2018

Links

Crossrefs

Cf. A018818, A074351, A074752.
Main diagonal of A256067, A256554.
Cf. A000837, A074761, A285572, A290103, A305566, A316429, A316431, A316432, A316433, A317624.

Programs

  • Maple
    A:= proc(n)
      uses numtheory;
      local S;
    S:= add(mobius(n/i)*1/mul(1-x^j,j=divisors(i)),i=divisors(n));
    coeff(series(S,x,n+1),x,n);
end proc:
seq(A(n),n=1..100); # Robert Israel, Aug 06 2014
  • Mathematica
    a[n_] := With[{s = Sum[MoebiusMu[n/i]*1/Product[1-x^j, {j, Divisors[i]}], {i, Divisors[n]}]}, SeriesCoefficient[s, {x, 0, n}]]; Array[a, 80] (* Jean-François Alcover, Feb 29 2016 *)
Table[Length[Select[IntegerPartitions[n],LCM@@#==n&]],{n,50}] (* Gus Wiseman, Aug 01 2018 *)
  • PARI
    pr(k, x)={my(t=1); fordiv(k, d, t *= (1-x^d) ); return(t); }
a(n) =
{
    my( x = 'x+O('x^(n+1)) );
    polcoeff( Pol( sumdiv(n, i, moebius(n/i) / pr(i, x) ) ), n );
}
vector(66, n, a(n) )
\\ Joerg Arndt, Aug 06 2014

Formula

Coefficient of x^n in expansion of Sum_{i divides n} A008683(n/i)*1/Product_{j divides i} (1-x^j).

A256067 Irregular table T(n,k): the number of partitions of n where the least common multiple of all parts equals k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 0, 0, 1, 0, 1, 1, 4, 2, 4, 1, 5, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 4, 3, 4, 1, 7, 1, 1, 1, 2, 0, 2, 0, 1, 1, 0, 0, 0, 0, 1, 1, 5, 3, 6, 2, 9, 1, 2, 1, 3, 0, 4, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 3, 6, 2
Offset: 0

Views

Author

R. J. Mathar, Mar 18 2015

Keywords

Examples

			The 5 partitions of n=4 are 1+1+1+1 (lcm=1), 1+1+2 (lcm=2), 2+2 (lcm=2), 1+3 (lcm=3) and 4 (lcm=4). So k=1, 3 and 4 appear once, k=2 appears twice.
The triangle starts:
  1 ;
  1 ;
  1  1;
  1  1  1;
  1  2  1  1;
  1  2  1  1  1  1;
  1  3  2  2  1  2;
  1  3  2  2  1  3  1  0  0  1  0  1;
  ...

Links

Crossrefs

Cf. A000041 (row sums), A000793 (row lengths), A213952, A074761 (diagonal), A074752 (6th column), A008642 (4th column), A002266 (5th column), A002264 (3rd column), A132270 (7th column), A008643 (8th column), A008649 (9th column), A258470 (10th column).
Cf. A009490 (number of nonzero terms of rows), A074064 (last elements of rows), A168532 (the same for gcd), A181844 (Sum k*T(n,k)).

Programs

  • Maple
    A256067 := proc(n,k)
        local a,p ;
        a := 0 ;
        for p in combinat[partition](n) do
                ilcm(op(p)) ;
                if % = k then
                        a := a+1 ;
                end if;
        end do:
        a;
end proc:
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x,
      b(n, i-1)+(p-> add(coeff(p, x, t)*x^ilcm(t, i),
      t=1..degree(p)))(add(b(n-i*j, i-1), j=1..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Mar 27 2015
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x, b[n, i-1] + Function[{p}, Sum[ Coefficient[p, x, t]*x^LCM[t, i], {t, 1, Exponent[p, x]}]][Sum[b[n-i*j, i-1], {j, 1, n/i}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jun 22 2015, after Alois P. Heinz *)

Extensions

T(0,1)=1 prepended by Alois P. Heinz, Mar 27 2015

A256554 Number T(n,k) of cycle types of degree-n permutations having the k-th smallest possible order; triangle T(n,k), n>=0, 1<=k<=A009490(n), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 1, 1, 1, 4, 2, 4, 1, 5, 1, 1, 1, 1, 1, 1, 4, 3, 4, 1, 7, 1, 1, 1, 2, 2, 1, 1, 1, 1, 5, 3, 6, 2, 9, 1, 2, 1, 3, 4, 1, 1, 1, 1, 1, 1, 5, 3, 6, 2, 12, 1, 2, 1, 4, 1, 6, 2, 2, 1, 2, 1, 1, 1, 2
Offset: 0

Views

Author

Alois P. Heinz, Apr 01 2015

Keywords

Comments

Sum_{k>=0} A256553(n,k)*T(n,k) = A181844(n).

Examples

			Triangle T(n,k) begins:
  1;
  1;
  1, 1;
  1, 1, 1;
  1, 2, 1, 1;
  1, 2, 1, 1, 1, 1;
  1, 3, 2, 2, 1, 2;
  1, 3, 2, 2, 1, 3, 1, 1, 1;
  1, 4, 2, 4, 1, 5, 1, 1, 1, 1, 1;
  1, 4, 3, 4, 1, 7, 1, 1, 1, 2, 2, 1, 1, 1;
  1, 5, 3, 6, 2, 9, 1, 2, 1, 3, 4, 1, 1, 1, 1, 1;

Links

Crossrefs

Row sums give A000041.
Row lengths give A009490.
Columns k=1-9 give: A000012, A004526, A002264, A008642(n-4), A002266, A074752, A132270, A008643(n-8) for n>7, A008649(n-9) for n>8.
Last elements of rows give A074064.
Main diagonal gives A074761.
Cf. A181844, A256067, A256553.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0 or i=1, x,
      b(n, i-1)+(p-> add(coeff(p, x, t)*x^ilcm(t, i),
      t=1..degree(p)))(add(b(n-i*j, i-1), j=1..n/i)))
    end:
T:= n->(p->seq((h->`if`(h=0, [][], h))(coeff(p, x, i))
     , i=1..degree(p)))(b(n$2)):
seq(T(n), n=0..12);
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x, b[n, i - 1] + Function[p, Sum[Coefficient[p, x, t]*x^LCM[t, i], {t, 1, Exponent[p, x]}]][Sum[b[n - i*j, i - 1], {j, 1, n/i}]]]; T[n_] := Function[p, Table[Function[h, If[h == 0, {{}, {}}, h]][Coefficient[p, x, i]], {i, 1, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 23 2017, translated from Maple *)
Showing 1-3 of 3 results.

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