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 1-4 of 4 results.

A008307 Table T(n,k) giving number of permutations of [1..n] with order dividing k, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 10, 3, 2, 1, 1, 26, 9, 4, 1, 1, 1, 76, 21, 16, 1, 2, 1, 1, 232, 81, 56, 1, 6, 1, 1, 1, 764, 351, 256, 25, 18, 1, 2, 1, 1, 2620, 1233, 1072, 145, 66, 1, 4, 1, 1, 1, 9496, 5769, 6224, 505, 396, 1, 16, 3, 2, 1, 1, 35696, 31041, 33616, 1345, 2052, 1, 56, 9, 4, 1, 1
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

Solutions to x^k = 1 in Symm_n (the symmetric group of degree n).

Examples

			Array begins:
  1,   1,    1,    1,    1,     1,    1,     1, ...
  1,   2,    1,    2,    1,     2,    1,     2, ...
  1,   4,    3,    4,    1,     6,    1,     4, ...
  1,  10,    9,   16,    1,    18,    1,    16, ...
  1,  26,   21,   56,   25,    66,    1,    56, ...
  1,  76,   81,  256,  145,   396,    1,   256, ...
  1, 232,  351, 1072,  505,  2052,  721,  1072, ...
  1, 764, 1233, 6224, 1345, 12636, 5761, 11264, ...

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 257.
  • J. D. Dixon, B. Mortimer, Permutation Groups, Springer (1996), Exercise 1.2.13.

Links

Crossrefs

Rows give A000034, A284517, A284518.
Columns give A000012, A000085, A001470, A001472, A052501, A053496, A053497, A053498, A053499, A053500, A053501, A053502, A053503, A053504, A053505.
Main diagonal gives A074759. - Alois P. Heinz, Feb 14 2013

Programs

  • Maple
    A:= proc(n,k) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..j-1)*A(n-j,k), j=numtheory[divisors](k))))
    end:
seq(seq(A(1+d-k, k), k=1..d), d=1..12); # Alois P. Heinz, Feb 14 2013
# alternative
A008307 := proc(n,m)
    local x,d ;
    add(x^d/d, d=numtheory[divisors](m)) ;
    exp(%) ;
    coeftayl(%,x=0,n) ;
    %*n! ;
end proc:
seq(seq(A008307(1+d-k,k),k=1..d),d=1..12) ; # R. J. Mathar, Apr 30 2017
  • Mathematica
    t[n_ /; n >= 0, k_ /; k >= 0] := t[n, k] = Sum[(n!/(n - d + 1)!)*t[n - d, k], {d, Divisors[k]}]; t[, ] = 1; Flatten[ Table[ t[n - k, k], {n, 0, 12}, {k, 1, n}]] (* Jean-François Alcover, Dec 12 2011, after given formula *)

Formula

T(n+1,k) = Sum_{d|k} (n)_(d-1)*T(n-d+1,k), where (n)_i = n!/(n - i)! = n*(n - 1)*(n - 2)*...*(n - i + 1) is the falling factorial.
E.g.f. for n-th row: Sum_{n>=0} T(n,k)*t^n/n! = exp(Sum_{d|k} t^d/d).

Extensions

More terms from Vladeta Jovovic, Apr 13 2001

A261431 Number of permutations p of [n] without fixed points such that p^n = Id.

Original entry on oeis.org

1, 0, 1, 2, 9, 24, 175, 720, 7665, 42560, 436401, 3628800, 70215145, 479001600, 7116730335, 88966701824, 1653438211425, 20922789888000, 457688776369825, 6402373705728000, 145083396337080201, 2457732174030848000, 55735573291977790575, 1124000727777607680000
Offset: 0

Views

Author

Alois P. Heinz, Aug 18 2015

Keywords

Links

Crossrefs

Main diagonal of A261430.
Cf. A074759.

Programs

  • Maple
    with(numtheory):
A:= proc(n, k) option remember; `if`(n<0, 0, `if`(n=0, 1,
      add(mul(n-i, i=1..j-1)*A(n-j, k), j=divisors(k) minus {1})))
    end:
a:= n-> A(n$2):
seq(a(n), n=0..25);
  • Mathematica
    A[n_, k_] := A[n, k] = If[n < 0, 0, If[n == 0, 1, Sum[Product[n - i, {i, 1, j - 1}] A[n - j, k], {j, Divisors[k] ~Complement~ {1}}]]];
a[n_] := A[n, n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 23 2017, translated from Maple *)

Formula

a(n) = n! * [x^n] exp(Sum_{d|n, d>1} x^d/d).

A335088 a(n) = n! * [x^n] exp(Sum_{k=1..n, gcd(n,k) = 1} x^k / k).

Original entry on oeis.org

1, 1, 1, 4, 9, 96, 145, 4320, 11025, 179200, 693441, 36288000, 62610625, 5748019200, 15259154625, 378002200576, 4108830350625, 334764638208000, 643373556450625, 115242726703104000, 309281197713168681, 14870534508544000000, 168541849294187630625
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 12 2020

Keywords

Comments

Number of permutations of [n] whose cycle lengths are relatively prime to n.

Links

Crossrefs

Cf. A074759, A226388, A335797.

Programs

  • Maple
    b:= proc(n, m) option remember; `if`(n=0, 1, add(`if`(
      igcd(j, m)=1, b(n-j, m), 0)*(n-1)!/(n-j)!, j=1..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Oct 12 2020
  • Mathematica
    Table[n! SeriesCoefficient[Exp[Sum[Boole[GCD[n, k] == 1] x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 22}]

A346121 Number of permutations of [n] whose order is a multiple of n.

Original entry on oeis.org

1, 1, 2, 6, 24, 240, 720, 5040, 40320, 816480, 3628800, 108108000, 479001600, 14789174400, 254431457280, 1307674368000, 20922789888000, 872545722048000, 6402373705728000, 411616608508385280, 7817896752906240000, 128126503414990080000, 1124000727777607680000
Offset: 1

Views

Author

Alois P. Heinz, Jul 05 2021

Keywords

Links

Crossrefs

Cf. A000142, A000961, A052699, A057731, A074351, A074759.

Programs

  • Maple
    b:= proc(n, g) option remember; `if`(n=0, x^g, add((j-1)!
      *b(n-j, ilcm(g, j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> (p-> add(coeff(p, x, i*n), i=1..degree(p)/n))(b(n, 1)):
seq(a(n), n=1..23);
# second Maple program:
h:= proc(n, j) option remember; uses padic, numtheory; n/mul(`if`(
      ordp(j, p) b(n$2):
seq(a(n), n=1..23);
  • Mathematica
    b[n_, g_] := b[n, g] = If[n == 0, x^g, Sum[(j - 1)!*b[n - j, LCM[g, j]]* Binomial[n - 1, j - 1], {j, 1, n}]] // Expand;
a[n_] := With[{p = b[n, 1]}, Sum[Coefficient[p, x, i*n], {i, 1, Exponent[p, x]/n}]];
Array[a, 40] (* Jean-François Alcover, Aug 23 2021, after Alois P. Heinz's first program *)

Formula

a(n) = Sum_{i>=1} A057731(n,i*n).
a(n) = (n-1)! <=> n in { A000961 }.
a(n) = A057731(n,n) = A074351(n) = A052699(n-1) for n <= 9.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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