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A297567
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Number of nonisomorphic proper colorings of partition star graph using three colors.
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8
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3, 6, 9, 12, 12, 24, 24, 15, 36, 30, 48, 48, 18, 48, 60, 72, 96, 96, 96, 21, 60, 90, 60, 96, 192, 108, 144, 192, 192, 192, 24, 72, 120, 120, 120, 288, 240, 216, 192, 384, 384, 288, 384, 384, 384, 27, 84, 150, 180, 105, 144, 384, 480, 324, 432, 240, 576, 480, 768, 408, 384, 768, 768, 576, 768, 768, 768, 30, 96, 180, 240, 210, 168, 480, 720, 480, 432, 864, 360, 288, 768, 960, 1152, 1536, 816, 480, 1152, 960, 1536, 1536, 768
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OFFSET
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0,1
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COMMENTS
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A partition star graph consists of a multiset of paths with lengths given by the elements of the partition attached to a distinguished root node. The ordering of the partitions is by traversing antichains in Young's lattice bottom to top, left to right. Isomorphism refers to the automorphisms of the star graph corresponding to the partition.
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LINKS
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FORMULA
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For a partition lambda we have the OCP: k Product_{p^v in lambda} C((k-1)^p+v-1, v). Here we have k=3.
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EXAMPLE
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Rows are:
3;
6;
9, 12;
12, 24, 24;
15, 36, 30, 48, 48;
18, 48, 60, 72, 96, 96, 96;
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MAPLE
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b:= (n, i)-> `if`(n=0, [3], `if`(i<1, [], [seq(map(x-> x*
binomial(2^i+j-1, j), b(n-i*j, i-1))[], j=0..n/i)])):
T:= n-> b(n$2)[]:
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MATHEMATICA
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b[n_, i_] := If[n == 0, {3}, If[i<1, {}, Table[Map[Function[x, x*Binomial[ 2^i + j - 1, j]], b[n - i*j, i - 1]], {j, 0, n/i}]] // Flatten];
T[n_] := b[n, n];
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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