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A276468 Irregular triangular array:  T(n,i) = number of partitions of n having crossover index k; see Comments. 3
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 (list; graph; refs; listen; history; text; internal format)
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: A112085 A090002 A061298 * A002126 A129721 A268193

Adjacent sequences:  A276465 A276466 A276467 * A276469 A276470 A276471

KEYWORD

nonn,easy,tabf

AUTHOR

Clark Kimberling, Dec 03 2016

STATUS

approved

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Last modified February 23 02:29 EST 2019. Contains 320411 sequences. (Running on oeis4.)