A276468 Irregular triangular array: T(n,i) = number of partitions of n having crossover index k; see Comments.

1, 2, 2, 1, 4, 1, 4, 2, 1, 7, 3, 1, 7, 6, 1, 1, 12, 8, 1, 1, 12, 12, 4, 1, 1, 19, 16, 5, 1, 1, 19, 25, 8, 2, 1, 1, 30, 34, 9, 2, 1, 1, 30, 44, 17, 6, 2, 1, 1, 45, 59, 20, 7, 2, 1, 1, 45, 81, 31, 12, 3, 2, 1, 1, 67, 108, 36, 13, 3, 2, 1, 1, 67, 132, 64, 18, 9

Offset: 1

Author

Clark Kimberling, Dec 03 2016

Keywords

• nonn
• easy
• tabf

Comments

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 n-th row sum is the number of partitions of n, A000041. The bisections of column 1 are given by A000070. The limit of the reversal of row n is given by A000041.

Examples

			First 15 rows (indexed by column 1):
1...   1
2...   2
3...   2    1
4...   4    1
5...   4    2     1
6...   7    3     1
7...   7    6     1     1
8...   12   8     1     1
9...   12   12    4     1     1
10..   19   16    5     1     1
11...  19   25    8     2     1    1
12..   30   34    9     2     1    1
13..   30   44    17    6     2    1    1
14..   45   59    20    7     2    1    1
15..   45   81    31    12    3    2    1    1
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A000041, A000070 (bisections of column 1), A279033 (crossover index for strict partitions), A279044 (crossover parts).

Programs

Mathematica

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[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, Ceiling[n/2]}];
TableForm[v] (* A276468 array *)
Flatten[v]   (* A276468 sequence *)