A279044 Irregular triangular array: T(n,i) = number of partitions of n having crossover part k; see Comments.

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

Offset

1,8

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 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.

Links

Clark Kimberling, Table of n, a(n) for n = 1..1000

Example

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

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[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}];
TableForm[v] (*A279044 array*)
Flatten[v]   (*A279044 sequence*)

Crossrefs

Cf. A000041, A276468 (crossover index array).

Sequence in context: A346080 A103921 A115623 * A134265 A182858 A175077

Adjacent sequences: A279041 A279042 A279043 * A279045 A279046 A279047

Keyword

nonn,easy,tabf

Author

Clark Kimberling, Dec 04 2016

Status

approved