A056376 -id:A056376 - 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

A065957 This is the case k = 3 of the number of orbits of the group of units of Z/(n) acting naturally on the k-subsets of Z/(n).

Original entry on oeis.org

1, 3, 3, 12, 7, 20, 16, 32, 17, 72, 25, 64, 65, 89, 43, 148, 55, 172, 123, 156, 81, 334, 118, 220, 175, 322, 131, 556, 151, 374, 291, 376, 289, 735, 217, 472, 405, 766, 267, 1028, 295, 760, 659, 692, 353, 1368, 446, 1008, 685, 1054, 451, 1484, 681, 1398, 855

Offset: 3

W. Edwin Clark, Jul 15 2001

nonn

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

Cf. A063379, A063381, A000005, A056376 + 1, A056371 - 1.

More terms from Sean A. Irvine, Apr 22 2023

A056386 Number of primitive (aperiodic) step shifted (decimated) sequences using exactly two different symbols.

Original entry on oeis.org

0, 2, 4, 8, 10, 32, 26, 84, 98, 266, 214, 1200, 702, 2770, 4328, 8832, 8230, 44086, 29202, 135744, 176720, 419654, 381490, 2149068, 1678244, 5593294, 7461064, 22550600, 19175158, 134386396, 71585134

Offset: 1

Marks R. Nester

nonn

See A056371 for an explanation of step shifts.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Sum mu(d)*A056376(n/d) where d divides n.

A056396 Number of step shifted (decimated) sequence structures using exactly two different symbols.

Original entry on oeis.org

0, 1, 2, 5, 5, 19, 13, 47, 51, 139, 107, 623, 351, 1399, 2171, 4463, 4115, 22111, 14601, 68015, 88375, 209935, 190745, 1075199, 839127, 2796999, 3730583, 11276703, 9587579, 67195519, 35792567

Offset: 1

Marks R. Nester

nonn

See A056371 for an explanation of step shifts. Permuting the symbols will not change the structure.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Column 2 of A288620.

Cf. A056376.

A056391(n)-1.

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

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A065957 This is the case k = 3 of the number of orbits of the group of units of Z/(n) acting naturally on the k-subsets of Z/(n).

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A056386 Number of primitive (aperiodic) step shifted (decimated) sequences using exactly two different symbols.

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Formula

A056396 Number of step shifted (decimated) sequence structures using exactly two different symbols.

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Author

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Formula