A065957 -id:A065957 - cp's OEIS frontend

A063379 Number of orbits of the group of units of Z/(n) acting naturally on the 2-subsets of Z/(n).

1, 2, 4, 3, 9, 4, 11, 8, 13, 6, 25, 7, 17, 18, 24, 9, 33, 10, 35, 23, 25, 12, 59, 18, 29, 26, 45, 15, 71, 16, 49, 33, 37, 32, 86, 19, 41, 38, 81, 21, 91, 22, 65, 61, 49, 24, 123, 32, 73, 48, 75, 27, 105, 46, 103, 53, 61, 30, 181, 31, 65, 78, 98, 53, 131, 34

Offset: 2

W. Edwin Clark, Jul 15 2001

nonn

			a(3) = 2 since when U(3) = {1,2} acts naturally on the three 2-subsets {0,1}, {0,2}, {1,2} of Z/(3) the orbits are {{0,1},{0,2}} and {{1,2}}. Note that 2{0,1} = {0,2} but there is no unit a in U(3) such that a{0,1} = {1,2}.

Cf. A065957, A063381, A000005, A056376 + 1, A056371 - 1

More terms from Sean A. Irvine, Apr 22 2023

A063381 Number of orbits of the group of units of Z/(n) acting naturally on the 4-subsets of Z/(n).

Original entry on oeis.org

1, 2, 9, 7, 24, 24, 56, 34, 151, 62, 173, 187, 264, 151, 530, 218, 679, 528, 737, 405, 1558, 638, 1256, 1002, 1871, 852, 3567, 1053, 2472, 2109, 2908, 2226, 5433, 1840, 4113, 3523, 6356, 2537, 9598, 2944, 7311, 6424, 7429, 3883, 13592, 5058, 11576, 7982, 12123, 5638, 17971, 8613, 16766, 11201

Offset: 4

W. Edwin Clark, Jul 15 2001

nonn

			a(5) = 2 since when U(5) = {1,2,3,4} acts naturally on the five 4-subsets {0,1,2,3}, {0,1,2,4}, {0,1,3,4}, {0,2,3,4}, {1,2,3,4} of Z/(5) the orbits are {{0,1,2,3},{0,1,2,4}, {0,1,3,4}, {0,2,3,4}} and {{1,2,3,4}}.

Cf. A063379, A065957, A000005, A056376 + 1, A056371 - 1

g:= proc(n) local U, S, C, R,u,s,Cr,us,v;
  U:= select(t -> igcd(n,t)=1, [$1..n-1]);
  S:= combinat:-choose({$0..n-1},4);
  C:= S;
  for s in S do R[s]:= s od;
  for u in U do
    Cr:= NULL;
    for s in C do
      us:= map(t -> u*t mod n, s);
      v:= R[us];
      while R[v] <> v do v:= R[v] od;
      if v <> s then R[s]:= v; Cr:= Cr, s fi
    od;
    C:= C minus {Cr};
  od;
  nops(C)
end proc;
map(g, [$4..60]); # Robert Israel, Nov 28 2022

Offset corrected, and more terms by Robert Israel, Nov 28 2022

cp's OEIS Frontend

A063379 Number of orbits of the group of units of Z/(n) acting naturally on the 2-subsets of Z/(n).

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