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 A368867 Number of labeled mappings from n points to themselves with unique square root (endofunctions). 1
 1, 1, 2, 2, 56, 544, 5064, 69348, 1210320 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS A mapping f has a unique square root if there exists a unique g such that gg = f. LINKS Table of n, a(n) for n=0..8. EXAMPLE For n = 3, the two 3-cycles are unique square roots of each other. Note that the identity map has more than one square root (i.e., 1->2, 2->1, 3->3 and itself). Another non-example: 1->1, 2->2, 3->1 has two square roots: itself and 1->2, 2->1, 3->2. In fact, the only endofunctions on {1,2,3} with unique square roots are the two 3-cycles, so a(3) = 2. PROG (Lua) function increment(size, t) t[1] = t[1] + 1 local index = 1 while t[index] > size do t[index] = 1 index = index + 1 if index > size then return true end t[index] = t[index] + 1 end return false end function get_initial(size) local return_value = {} for i = 1, size do return_value[i] = 1 end return return_value end function compute(size) candidate = get_initial(size) return_value = 0 repeat fun_root = get_initial(size) fun_root_count = 0 repeat for i = 1, size do if candidate[i] ~= fun_root[fun_root[i]] then goto next_fun_root end end fun_root_count = fun_root_count + 1 if (fun_root_count == 2) then break end ::next_fun_root:: until (increment(size, fun_root)) if (fun_root_count == 1) then return_value = return_value + 1 end until (increment(size, candidate)) return return_value end CROSSREFS A088994 (permutations only) < This sequence < A102687 (any square maps) < A000312 (all maps). Sequence in context: A323872 A037292 A239151 * A286377 A187024 A274477 Adjacent sequences: A368864 A368865 A368866 * A368868 A368869 A368870 KEYWORD nonn,hard,more AUTHOR Keith J. Bauer, Jan 08 2024 EXTENSIONS a(7)-a(8) from Andrew Howroyd, Jan 09 2024 STATUS approved

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Last modified September 8 08:54 EDT 2024. Contains 375753 sequences. (Running on oeis4.)