A394566 Number of labeled elementary Abelian groups (C_2)^n with a fixed identity.

1, 1, 30, 64864800, 822336494953469303808000000, 98349960836205821940270184651038804258510922190452164539040399360000000000000

Offset

1,3

Comments

Equivalent to number of reduced Latin squares representing (C_2^n).

Equivalent to number of maximal classes determined by elementary Abelian p-groups of order 2^n given by A246137(2^n).

Links

Table of n, a(n) for n=1..6.

Ivo Rosenberg, The number of maximal closed classes in the set of functions over a finite domain, J. Combin. Theory Ser. A 14 (1973), 1-7.

Matt Samuel, XOR is commutative, associative, and its own inverse. Are there any other such functions?, MathStackExchange.

Formula

a(n) = (2^n-1)! / A002884(n).

a(n) = (2^n-1)! / (A006125(n) * A005329(n)).

a(n) = (2^n-1)! / |Aut((C_2)^n)|.

a(n) = A246137(2^n).

a(k+1) = a(k) * A001761(2^k-1), for k>0.

Example

1 2 3 4 5 6 7 8      1 2 3 4 5 6 7 8
2 1 4 3 6 5 8 7      2 1 5 6 3 4 8 7
3 4 1 2 7 8 5 6      3 5 1 7 2 8 4 6
4 3 2 1 8 7 6 5      4 6 7 1 8 2 3 5
5 6 7 8 1 2 3 4      5 3 2 8 1 7 6 4
6 5 8 7 2 1 4 3      6 4 8 2 7 1 5 3
7 8 5 6 3 4 1 2      7 8 4 3 6 5 1 2
8 7 6 5 4 3 2 1      8 7 6 5 4 3 2 1
----------
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
2 1 4 3 6 5 8 7 10 9 12 11 14 13 16 15
3 4 1 2 7 8 5 6 11 12 9 10 15 16 13 14
4 3 2 1 8 7 6 5 12 11 10 9 16 15 14 13
5 6 7 8 1 2 3 4 13 14 15 16 9 10 11 12
6 5 8 7 2 1 4 3 14 13 16 15 10 9 12 11
7 8 5 6 3 4 1 2 15 16 13 14 11 12 9 10
8 7 6 5 4 3 2 1 16 15 14 13 12 11 10 9
9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8
10 9 12 11 14 13 16 15 2 1 4 3 6 5 8 7
11 12 9 10 15 16 13 14 3 4 1 2 7 8 5 6
12 11 10 9 16 15 14 13 4 3 2 1 8 7 6 5
13 14 15 16 9 10 11 12 5 6 7 8 1 2 3 4
14 13 16 15 10 9 12 11 6 5 8 7 2 1 4 3
15 16 13 14 11 12 9 10 7 8 5 6 3 4 1 2
16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
----------
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
2 1 4 3 6 5 8 7 10 9 12 11 14 13 16 15
3 4 1 2 8 7 6 5 12 11 10 9 15 16 13 14
4 3 2 1 7 8 5 6 11 12 9 10 16 15 14 13
5 6 8 7 1 2 4 3 14 13 15 16 10 9 11 12
6 5 7 8 2 1 3 4 13 14 16 15 9 10 12 11
7 8 6 5 4 3 1 2 15 16 14 13 12 11 9 10
8 7 5 6 3 4 2 1 16 15 13 14 11 12 10 9
9 10 12 11 14 13 15 16 1 2 4 3 6 5 7 8
10 9 11 12 13 14 16 15 2 1 3 4 5 6 8 7
11 12 10 9 15 16 14 13 4 3 1 2 8 7 5 6
12 11 9 10 16 15 13 14 3 4 2 1 7 8 6 5
13 14 15 16 10 9 12 11 6 5 8 7 1 2 3 4
14 13 16 15 9 10 11 12 5 6 7 8 2 1 4 3
15 16 13 14 11 12 9 10 7 8 5 6 3 4 1 2
16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Crossrefs

Cf. A336412 (reduced Latin squares representing dihedral groups).

Cf. A058161 (reduced Latin squares representing cyclic groups).

Cf. A246137 (reduced Latin squares representing elementary Abelian p-groups).

Cf. A000315 (reduced Latin squares).

Cf. A005154 (Latin stable marriage problem, power-of-2 orders).

Cf. A002884 (the number of automorphisms of (C_2)^n).

Cf. A006125, A005329, A001761.

Sequence in context: A140762 A028363 A382282 * A091511 A324684 A213070

Adjacent sequences: A394563 A394564 A394565 * A394567 A394568 A394569

Keyword

nonn,easy

Author

Dan Eilers, Mar 24 2026

Status

approved