A215651 Number of transformation semigroups acting on n points (counting conjugates as one), i.e., the number of subsemigroups of the full transformation semigroup T_n.

1, 2, 8, 283, 132069776

Offset: 0

Author

Attila Egri-Nagy, Aug 19 2012

Keywords

• nonn
• more

Comments

The semigroup analog of A000638.

We apply the categorical viewpoint and consider the empty set as a semigroup.

Links

• James East, Attila Egri-Nagy, James D. Mitchell, Enumerating Transformation Semigroups, Semigroup Forum 95, 109-125 (2017); arXiv: 1403.0274 [math.GR], 2014-2017.

Crossrefs

Cf. A000638, A215650.

Programs

GAP

################################################################################
# GAP 4.5 function calculating the conjugacy classes of a set of subsemigrops.
# (C) 2012 Attila Egri-Nagy www.egri-nagy.hu
# GAP can be obtained from www.gap-system.org
################################################################################
# Input: list of subsemigroups of a transformation semigroup,
#        automorphism group of the semigroup
# Output: list of conjugacy classes
ConjugacyClassesSubsemigroups := function(subsemigroups, G)
local ssg, #subsemigroup
      ccl, #conjugacy class
      ccls; #result: all conjugacy classes
  ccls := [];
  for ssg in subsemigroups do
    #we check whether the subsemigroup is already in a conjugacy class
    if not ForAny(ccls, x -> ssg in x) then
      #conjugating by all group elements
      ccl := DuplicateFreeList(
                     List(G,
                          g -> AsSortedList(List(ssg, t-> t^g))));
      Add(ccls, ccl);
    fi;
  od;
  return ccls;
end;

Extensions

a(4) moved from a comment by Attila Egri-Nagy, Jan 09 2014 to data by Andrey Zabolotskiy, Mar 25 2021