A378280 -id:A378280 - cp's OEIS frontend

A378163 Triangle read by rows: T(n,k) is the number of subgroups of S_n isomorphic to S_k, where S_n is the n-th symmetric group.

1, 1, 1, 1, 3, 1, 1, 9, 4, 1, 1, 25, 20, 5, 1, 1, 75, 160, 60, 12, 1, 1, 231, 910, 560, 84, 7, 1, 1, 763, 5936, 5740, 560, 56, 8, 1, 1, 2619, 53424, 58716, 3276, 336, 72, 9, 1, 1, 9495, 397440, 734160, 79632, 4620, 480, 90, 10, 1, 1, 35695, 3304620, 8337120, 1105104, 39732, 3300, 660, 110, 11, 1, 1, 140151, 35023120, 133212420, 16571808, 1400784, 20592, 4950, 880, 132, 12, 1, 1, 568503, 322852816, 1769490580, 176344740, 16253952, 130416, 33462, 7150, 1144, 156, 13, 1

Offset: 1

Jianing Song, Nov 18 2024

The number of monomorphisms (i.e., injective homomorphisms) S_k -> S_n is thus |Aut(S_k)|*T(n,k). Note that |Aut(S_k)| = 1 for k = 2, 1440 for k = 6 and k! otherwise.
T(n,k) is related to the number of homomorphisms S_k -> S_n:
     k     | trivial kernel | kernel S_k (k>=2) | kernel A_k (k>=3) | kernel V (k=4) |       total number
-----------+----------------+-------------------+-------------------+----------------+-------------------------
     1     |        1       |         -         |         -         |        -       |            1
-----------+----------------+-------------------+-------------------+----------------+-------------------------
     2     |     b(n)-1     |         1         |         -         |        -       |           b(n)
-----------+----------------+-------------------+-------------------+----------------+-------------------------
     4     |    24*T(n,4)   |         1         |       b(n)-1      |    6*T(n,3)    | 24*T(n,4)+6*T(n,3)+b(n)
-----------+----------------+-------------------+-------------------+----------------+-------------------------
     6     |   1440*T(n,6)  |         1         |       b(n)-1      |         -      |     1440*T(n,6)+b(n)
-----------+----------------+-------------------+-------------------+----------------+-------------------------
 3, 5, >=7 |    k!*T(n,k)   |         1         |       b(n)-1      |         -      |      k!*T(n,k)+b(n)
Here A_n is the n-th alternating group, V = {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} is the Klein-four group in S_4, b = A000085, and T(n,k) = 0 for k > n.
In particular, the number of homomorphisms S_n -> S_n is 1 for n = 1, 2 for n = 2, 58 for n = 4, 1440 + b(6) = 1516 for n = 6, and n! + b(n) otherwise.

			Table reads
  1
  1, 1
  1, 3, 1
  1, 9, 4, 1
  1, 25, 20, 5, 1
  1, 75, 160, 60, 12, 1
  1, 231, 910, 560, 84, 7, 1
  1, 763, 5936, 5740, 560, 56, 8, 1
  1, 2619, 53424, 58716, 3276, 336, 72, 9, 1
  1, 9495, 397440, 734160, 79632, 4620, 480, 90, 10, 1

Mathematics Stack Exchange, Subgroups in GAP

Cf. A000085, A001189, A378279, A378280, A378281, A281097.

A378163 := function(n,k)
local S;
S := SymmetricGroup(n);
return Sum(IsomorphicSubgroups(S,SymmetricGroup(k)),x->Index(S,Normalizer(S,Image(x))));
end; # program given in the Math Stack Exchange link

A378163_row_n := function(n)
local L, C, G, N, k;
N := ListWithIdenticalEntries( n, 0 );
L := ConjugacyClassesSubgroups( SymmetricGroup(n) );
for C in L do
G := Representative(C);
for k in [1..n] do
if not IsomorphismGroups( G, SymmetricGroup(k) ) = fail then
N[k] := N[k]+Size(C);
fi;
od;
od;
return N;
end;

T(n,2) = A001189(n).

A378279 Number of subgroups of S_n isomorphic to S_3, where S_n is the n-th symmetric group.

Original entry on oeis.org

0, 0, 1, 4, 20, 160, 910, 5936, 53424, 397440, 3304620, 35023120, 322852816

Offset: 1

Jianing Song, Nov 21 2024

Column k=3 of A378163.

Cf. A000085, A378280, A378281, A281097.

A378279 := function(n)
local S;
S := SymmetricGroup(n);
return Sum(IsomorphicSubgroups(S, SymmetricGroup(3)), x->Index(S, Normalizer(S, Image(x))));
end;

a(n) = (A281097(n) - A000085(n))/6. See A378163 for more information.

A378281 Number of subgroups of S_n isomorphic to S_5, where S_n is the n-th symmetric group.

Original entry on oeis.org

0, 0, 0, 0, 1, 12, 84, 560, 3276, 79632, 1105104, 16571808, 176344740

Offset: 1

Jianing Song, Nov 21 2024

Column k=5 of A378163.

Cf. A000085, A378279, A378280.

A378281 := function(n)
local S;
S := SymmetricGroup(n);
return Sum(IsomorphicSubgroups(S, SymmetricGroup(5)), x->Index(S, Normalizer(S, Image(x))));
end;

cp's OEIS Frontend

A378163 Triangle read by rows: T(n,k) is the number of subgroups of S_n isomorphic to S_k, where S_n is the n-th symmetric group.

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A378279 Number of subgroups of S_n isomorphic to S_3, where S_n is the n-th symmetric group.

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A378281 Number of subgroups of S_n isomorphic to S_5, where S_n is the n-th symmetric group.

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