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

1,2

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.

Links

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A353843 A090002 A061298 * A002126 A350815 A129721

Adjacent sequences: A276465 A276466 A276467 * A276469 A276470 A276471

Keyword

nonn,easy,tabf

Author

Clark Kimberling, Dec 03 2016

Status

approved