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A183568
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Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the number of partitions of n containing a clique of size k.
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22
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1, 1, 1, 2, 1, 1, 3, 2, 0, 1, 5, 3, 2, 0, 1, 7, 6, 2, 1, 0, 1, 11, 7, 3, 2, 1, 0, 1, 15, 13, 5, 3, 1, 1, 0, 1, 22, 16, 9, 3, 3, 1, 1, 0, 1, 30, 25, 10, 6, 3, 2, 1, 1, 0, 1, 42, 33, 16, 8, 5, 3, 2, 1, 1, 0, 1, 56, 49, 23, 13, 6, 5, 2, 2, 1, 1, 0, 1, 77, 61, 31, 15, 10, 5, 5, 2, 2, 1, 1, 0, 1
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OFFSET
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0,4
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COMMENTS
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All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique. Each partition has a clique of size 0.
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LINKS
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FORMULA
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G.f. of column k: (1-Product_{j>0} (1-x^(k*j)+x^((k+1)*j))) / (Product_{j>0} (1-x^j)).
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EXAMPLE
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T(5,2) = 2, because 2 (of 7) partitions of 5 contain (at least) one clique of size 2: [1,2,2], [1,1,3].
Triangle T(n,k) begins:
1;
1, 1;
2, 1, 1;
3, 2, 0, 1;
5, 3, 2, 0, 1;
7, 6, 2, 1, 0, 1;
11, 7, 3, 2, 1, 0, 1;
15, 13, 5, 3, 1, 1, 0, 1;
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MAPLE
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b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
add((l->`if`(j=k, [l[1]$2], l))(b(n-i*j, i-1, k)), j=0..n/i)))
end:
T:= (n, k)-> (l-> l[`if`(k=0, 1, 2)])(b(n, n, k)):
seq(seq(T(n, k), k=0..n), n=0..12);
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MATHEMATICA
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b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == k, {l[[1]], l[[1]]}, l]][b[n - i*j, i-1, k]], {j, 0, n/i}]] ]; t[n_, k_] := Function[l, l[[If[k == 0, 1, 2]]]][b[n, n, k]]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 16 2013, translated from Maple *)
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CROSSREFS
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Columns k=0-10 give: A000041, A183558, A183559, A183560, A183561, A183562, A183563, A183564, A183565, A183566, A183567.
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KEYWORD
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AUTHOR
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STATUS
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approved
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