A181964
Sum of the sizes of normalizers of all the cyclic subgroups of Alternating Group of order n.
Original entry on oeis.org
1, 1, 6, 36, 240, 2160, 20160, 241920, 2903040, 39916800, 578793600, 9580032000, 161902540800, 3007651046400, 58845346560000, 1234444603392000, 26854400821248000, 624231436308480000, 15083992450695168000, 385614968295997440000
Offset: 1
Decomposing by number of cyclic subgroups * size of normalizer of subgroups
a(5) = 1*60 + 4*15 + 6*10 + 0*60 + 10*6 = 240.
a(6) = 1*360 + 8*45 + (18*20+18*20) + 8*45 + 10*36 = 2160.
A244524
Number of pairs (f,g) of commuting maps {0,..,n-1}->{0,..,n-1} with 0 <= f(k), g(k) <= k.
Original entry on oeis.org
1, 1, 4, 26, 236, 2780, 40642, 715836, 14873174, 358866952, 9934283924, 312461402424
Offset: 0
The a(3) = 26 pairs of such maps are (dots for zeros in the maps):
01: [ . . . ] [ . . . ]
02: [ . . . ] [ . . 1 ]
03: [ . . . ] [ . . 2 ]
04: [ . . . ] [ . 1 . ]
05: [ . . . ] [ . 1 1 ]
06: [ . . . ] [ . 1 2 ]
07: [ . . 1 ] [ . . . ]
08: [ . . 1 ] [ . . 1 ]
09: [ . . 1 ] [ . 1 2 ]
10: [ . . 2 ] [ . . . ]
11: [ . . 2 ] [ . . 2 ]
12: [ . . 2 ] [ . 1 . ]
13: [ . . 2 ] [ . 1 2 ]
14: [ . 1 . ] [ . . . ]
15: [ . 1 . ] [ . . 2 ]
16: [ . 1 . ] [ . 1 . ]
17: [ . 1 . ] [ . 1 2 ]
18: [ . 1 1 ] [ . . . ]
19: [ . 1 1 ] [ . 1 1 ]
20: [ . 1 1 ] [ . 1 2 ]
21: [ . 1 2 ] [ . . . ]
22: [ . 1 2 ] [ . . 1 ]
23: [ . 1 2 ] [ . . 2 ]
24: [ . 1 2 ] [ . 1 . ]
25: [ . 1 2 ] [ . 1 1 ]
26: [ . 1 2 ] [ . 1 2 ]
Cf.
A181162 (commuting maps {1,..,n}->{1,..,n} without restrictions).
Cf.
A053529 (commuting permutations).
-
s:= proc(n) option remember; `if`(n=0, [[]],
map(x-> seq([x[], i], i=1..n), s(n-1)))
end:
a:= n-> (l-> add(add(`if`([seq(evalb(f[g[i]]=g[f[i]])
, i=1..n)]=[true$n], 1, 0), g=l), f=l))(s(n)):
seq(a(n), n=0..6); # Alois P. Heinz, Jul 30 2014
A350225
Number of ordered pairs (a,g) with a in IS_n the symmetric inverse semigroup on [n] and g in symmetric group on [n] such that ag=ga.
Original entry on oeis.org
1, 2, 10, 60, 480, 4320, 46800, 554400, 7459200, 108864000, 1745452800, 30017433600, 558036864000, 11021826816000, 232330146048000, 5173159799808000, 121812482727936000, 3012672515973120000, 78301030421053440000, 2127572806150471680000, 60438151687124090880000
Offset: 0
-
a:= proc(n) option remember; `if`(n=0, 1, 2*(n-1)!*
add(a(j)/j!*numtheory[sigma](n-j), j=0..n-1))
end:
seq(a(n), n=0..20); # Alois P. Heinz, Dec 20 2021
-
nn = 16; Table[Sum[PartitionsP[k] PartitionsP[n - k], {k, 0, n}], {n, 0, nn}] Table[n!, {n, 0, nn}]
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