A218915 Number of missing subgroup orders of the symmetric group, that is, i divides Factorial(n) but the symmetric group on n points does not have a subgroup of order i.

0, 0, 0, 0, 0, 3, 9, 29, 47, 86, 157, 401, 576, 1316

Offset: 0

Author

Liam Naughton, Nov 09 2012

Keywords

• nonn
• more

Links

• L. Naughton and G. Pfeiffer, Integer sequences realized by the subgroup pattern of the symmetric group, arXiv:1211.1911 [math.GR], 2012-2013.
• Liam Naughton, CountingSubgroups.g
• Liam Naughton and Goetz Pfeiffer, Tomlib, The GAP table of marks library

Programs

GAP

Size(Difference(DivisorsInt(Factorial(n)), DuplicateFreeList(List(ConjugacyClassesSubgroups(SymmetricGroup(n)), x->Size(Representative(x))))));

Mathematica

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A027423 = A@027423;
A218913 = A@218913;
a[n_] := A027423[[n+1]] - A218913[[n+1]];
a /@ Range[0, 13] (* Jean-François Alcover, Jan 08 2020 *)

Formula

a(n) = A027423(n) - A218913(n).