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 A215650 Number of transformation semigroups acting on n points (counting conjugates as distinct); also the number of subsemigroups of the full transformation semigroup T_n. 1
 1, 2, 10, 1299, 3161965550 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The semigroup analog of A005432. We apply the categorical viewpoint and consider the empty set as a semigroup. The first 4 terms can be calculated by brute force search (see attached program). LINKS Table of n, a(n) for n=0..4. James East, Attila Egri-Nagy, James D. Mitchell, Enumerating Transformation Semigroups, Semigroup Forum 95, 109-125 (2017); arXiv: 1403.0274 [math.GR], 2014-2017. PROG (GAP) ################################################################################ # GAP 4.5 function implementing a brute force search for submagmas of a magma. # (C) 2012 Attila Egri-Nagy www.egri-nagy.hu # GAP can be obtained from www.gap-system.org ################################################################################ # The function goes through all the subsets of the given magma (groups, # semigroups) and checks whether they form a magma or not. # If yes, then the submagma is collected. # The function returns the list of all (nonempty) submagmas. BruteForceSubMagmaSearch := function(M) local bitlist, #the characteristic function of a subset i, #an integer to index through the bitlist n, #size of the input magma elms, #elements of the magma gens, #generator set of a submagma submagmas, #the submagmas duplicates, #for counting how many times we encounter the same submagma nonsubmagmas; #counting how many subsets are not submagmas # duplicates + nonsubmagmas = 2^n-1 n := Size(M); submagmas := []; elms := AsList(M); duplicates := 0; nonsubmagmas := 0; bitlist := BlistList([1..n], [1]); #we start with the first element, the #empty set can be added afterwards, if the magma's definition allows it repeat #constructing a generator set based on the bitlist########################## gens := []; Perform([1..n], function(x) if bitlist[x] then Add(gens, elms[x]); fi; end); #checking whether it is a submagma if Size(gens) = Size(Magma(gens)) then if gens in submagmas then duplicates := duplicates + 1; else AddSet(submagmas, gens); fi; else nonsubmagmas := nonsubmagmas + 1; fi; #binary +1 applied to bitlist############################################### i := 1; while (i<=n) and (bitlist[i]) do bitlist[i] := false; i := i + 1; od; if i <= n then bitlist[i] := true; fi; ############################################################################ until SizeBlist(bitlist) = 0; Print("#I Submagmas:", Size(submagmas), " Duplicates:", duplicates, " Nonsubmagmas:", nonsubmagmas, "\n"); return submagmas; end; CROSSREFS Cf. A005432, A215651. Sequence in context: A245728 A171485 A291882 * A057015 A059732 A334575 Adjacent sequences: A215647 A215648 A215649 * A215651 A215652 A215653 KEYWORD nonn,hard,more,nice AUTHOR Attila Egri-Nagy, Aug 19 2012 EXTENSIONS a(4) moved from comments to data by Andrey Zabolotskiy, Mar 25 2021 STATUS approved

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Last modified March 1 06:29 EST 2024. Contains 370430 sequences. (Running on oeis4.)