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 A241900 Irregular triangular array:  T(n,k) = number of partitions (nodes) in the k-th component of the graph G'(n) obtained from the partition graph G(n) by deleting all partitions having repeated parts; G and G' are defined in Comments. 2
 1, 1, 2, 2, 2, 1, 2, 2, 3, 2, 4, 2, 5, 2, 1, 6, 2, 2, 7, 3, 2, 8, 5, 2, 9, 7, 2, 10, 9, 2, 1, 12, 11, 2, 2, 15, 12, 3, 2, 18, 13, 5, 2, 22, 14, 8, 2, 26, 15, 11, 2, 30, 16, 15, 2, 1, 35, 20, 17, 2, 2, 40, 26, 18, 3, 2, 45, 33, 19, 5, 2, 51, 41, 20, 8, 2, 57 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The partition graph G(n) is defined at A241150 as follows:  the nodes are the partitions of n, and nodes p and q have an edge if one of them can be obtained from the other by a substitution x -> x-1,1 for some part x.   Let R be the set of partitions (nodes) of n that contain a repeated part and let E be the set of edges of G(n) that have a node in R.  Removing R and E from G(n) leaves a graph G'(n) whose nodes are the strict partitions of n, as in A000009.  For n >= 5, G'(n) is disconnected. LINKS EXAMPLE The first 18 rows of T are represented here: row 1:  1 row 2:  1 row 3:  2 row 4:  2 row 5:  2 1 row 6:  2 2 row 7:  3 2 row 8:  4 2 row 9:  5 2 1 row 10:  6 2 2 row 11:  7 3 2 row 12:  8 5 2 row 13:  9 7 2 row 14:  10 9 2 1 row 15:  12 11 2 2 row 16:  15 12 3 2 row 17:  18 13 5 2 row 18:  22 14 8 2 The 10 nodes and 7 edges of G'(10) are shown here:  [10] - [9,1], [8,2] - [7,2,1], [7,3] - [6,3,1], [7,3] - [7,2,1], [6,4] - [5,4,1], [6,4] - [6,3,1], [5,3,2] - [4,3,2,1]; the three components are as follows:  [8,2] - [7,2,1] - [7,3] - [6,3,1] - [6,4] - [5,4,1]  (6 nodes); [4,3,2,1] - [5,3,2] (2 nodes); [9,1] - [10]] (2 nodes).  Thus, row 10 of the array is 6 2 2. MATHEMATICA (* The first program generates terms of A241900 and A241901. *) z = 30;  spawn[part_] := Map[Reverse[Sort[Flatten[ReplacePart[part, {# - 1, 1}, Position[part, #, 1, 1][[1]][[1]]]]]] &, DeleteCases[DeleteDuplicates[part], 1]]; findComponent[start_] := Reap[BreadthFirstScan[g, start, {"DiscoverVertex" -> ((PropertyValue[{g, #1}, "Visited"] = True; Sow[#1]) &)}]][[2, 1]]; subGLengths = Join[{{1}}, Table[parts = Select[IntegerPartitions[k], DeleteDuplicates[#] == # &]; graph = Flatten[Table[part = parts[[n]]; Map[{part, #} &, Select[spawn[part], DeleteDuplicates[#] == # &]], {n, 1, Length[parts]}], 1]; isolated = Map[{#, #} &, Map[#[[1]] &, Cases[Map[{#, MemberQ[Flatten[graph, 1], #]} &, parts], {{___}, False}]]]; graph = Join[graph, isolated]; {graph, isolated} = Map[Map[FromDigits[#[[1]]] <-> FromDigits[#[[2]]] &, #] &, {graph, isolated}]; g = Graph[graph]; Do[PropertyValue[{g, v}, "Visited"] = False, {v, VertexList[g]}]; vlists = Reap[Do[If[! PropertyValue[{g, start}, "Visited"], Sow[findComponent[start]]], {start, VertexList[g]}]][[2, 1]]; Reverse[Sort[Map[Length, vlists]]], {k, 2, z}]]; Flatten[%] (* A241900 *) Map[#[[1]] &, subGLengths] (* A241901, Peter J. C. Moses, Apr 30 2014 *) (* The next program shows the graph G'(z) for user-chosen z. *) z = 18; spawn[part_] := Map[Reverse[Sort[Flatten[ReplacePart[part, {# - 1, 1}, Position[part, #, 1, 1][[1]][[1]]]]]] &, DeleteCases[DeleteDuplicates[part], 1]]; findComponent[start_] := Reap[BreadthFirstScan[g, start, {"DiscoverVertex" -> ((PropertyValue[{g, #1}, "Visited"] = True; Sow[#1]) &)}]][[2, 1]]; parts = Select[IntegerPartitions[z], DeleteDuplicates[#] == # &]; graph = Flatten[Table[part = parts[[n]]; Map[{part, #} &, Select[spawn[part], DeleteDuplicates[#] == # &]], {n, 1, Length[parts]}], 1]; isolated = Map[{#, #} &, Map[#[[1]] &, Cases[Map[{#, MemberQ[Flatten[graph, 1], #]} &, parts], {{___}, False}]]]; graph = Join[graph, isolated]; graph = Map[FromDigits[#[[1]]] <-> FromDigits[#[[2]]] &, graph]; isolated = Map[FromDigits[#[[1]]] <-> FromDigits[#[[2]]] &, isolated]; g = Graph[graph, VertexLabels -> "Name", ImageSize -> 500, ImagePadding -> 20, If[Length[isolated] > 0, Apply[EdgeStyle -> {# -> White} &, isolated], EdgeStyle -> "Default"], GraphLayout -> "SpringElectricalEmbedding"] (* Peter J. C. Moses, Apr 30 2014 *) CROSSREFS Cf. A241901, A241150, A000041, A000009. Sequence in context: A189684 A106493 A083338 * A204018 A261915 A109037 Adjacent sequences:  A241897 A241898 A241899 * A241901 A241902 A241903 KEYWORD nonn,easy,tabf AUTHOR Clark Kimberling, May 01 2014 STATUS approved

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