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A279044
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Irregular triangular array: T(n,i) = number of partitions of n having crossover part k; see Comments.
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2
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1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 3, 2, 1, 1, 3, 4, 1, 3, 2, 1, 1, 3, 5, 2, 5, 3, 2, 1, 1, 5, 7, 5, 1, 5, 3, 2, 1, 1, 5, 9, 7, 2, 7, 5, 3, 2, 1, 1, 7, 12, 12, 5, 1, 7, 5, 3, 2, 1, 1, 7, 14, 16, 8, 2, 11, 7, 5, 3, 2, 1, 1, 11, 20, 20, 14, 5
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OFFSET
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1,8
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COMMENTS
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Suppose that P = [p(1),p(2),...,p(k)] is a partition of n, where p(1) >= p(2) >= ... >= p(k). The crossover index of P is the least h such that p(1) + ... + p(h) > = n/2. Equivalently for k > 1, p(1) + ... + p(h) >= p(h+1) + ... + p(k). The crossover part of P is p(h). The n-th row sum is the number of partitions of n, A000041. The bisections of column 1 are also given by A000041. The limit of the reversal of row n is given by A000041.
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LINKS
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EXAMPLE
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First 14 rows (indexed by column 1):
1... 1
2... 1 1
3... 1 1 1
4... 1 2 1 1
5... 2 1 2 1 1
6... 2 2 3 2 1 1
7... 3 4 1 3 2 1 1
8... 3 5 2 5 3 2 1 1
9... 5 7 5 1 5 3 2 1 1
10... 5 9 7 2 7 5 3 2 1 1
11... 7 12 12 5 1 7 5 4 2 1 1
12... 7 14 15 8 2 11 7 5 3 2 1 1
13... 11 20 20 14 5 1 11 7 5 3 2 1 1
14... 11 24 25 20 8 2 15 11 7 5 3 2 1 1
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MATHEMATICA
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p[n_] := p[n] = IntegerPartitions[n]; t[n_, k_] := t[n, k] = p[n][[k]];
q[n_, k_] := q[n, k] = Select[Range[50], Sum[t[n, k][[i]], {i, 1, #}] >= n/2 &, 1];
u[n_] := u[n] = Flatten[Table[p[n][[k]][[q[n, k]]], {k, 1, Length[p[n]]}]];
c[n_, k_] := c[n, k] = Count[u[n], k];
v = Table[c[n, k], {n, 1, 25}, {k, 1, n}];
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CROSSREFS
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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STATUS
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approved
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