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Comments from Thomas Wieder (wieder.thomas(AT)t-online.de), Oct 21 2004: "Also the number of hierarchies with labeled elements and labeled levels where the levels are permuted. Let l_x denote level x, e.g. l_2 is level 2. Let 1 denote an element and 2 a second element and so on. Then l_1:123 means elements 1,2 and 3 are on level 1.
"Let | indicate separation between levels. Then l_1:1|l_2:346|l_3:5 denotes a hierarchy of n=6 unlabeled elements with element 1 on level 1, elements 3,4 and 6 on level 2 and element 5 on level 3.
"E.g. for n=3 one has a(3) = 49 possible hierarchies:
"l_1:123,
"l_1:12|l_2:3, l_1:13|l_2:2, l_1:23|l_2:1,
"l_2:12|l_1:3, l_2:13|l_1:2, l_2:23|l_1:1,
"l_1:1|l_2:23, l_1:2|l_2:13, l_1:3|l_2:12,
"l_2:1|l_1:23, l_2:2|l_1:13, l_2:3|l_1:12,
"l_1:1|l_2:2|l_3:3 and further five permutations of the elements with levels fixed,
"l_3:1|l_1:2|l_2:3 and further five permutations of the elements with levels fixed,. etc., up to
"l_3:1|l_2:2|l_1:3 and further five permutations of the elements with levels fixed. this gives 1 + 6 +6 + 6*6 = 49 = a(3) possible hierarchies.
"See A001339 for the number of hierarchies with unlabeled elements and labeled levels."
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