OFFSET
1,3
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). A strict partition is a partition into distinct parts. The n-th row sum is the number of strict partitions of n, A000009. Column 1 counts "non-squashing partitions", as in A088567.
First 32 rows (indexed by column 1):
1... 1
2... 1
3... 2
4... 2
5... 3
6... 4
7... 5
8... 6
9... 7 1
10... 9 1
11... 10 2
12... 13 2
13... 14 4
14... 18 4
15... 19 8
16... 24 8
17... 25 13
18... 32 14
19... 33 21
20... 42 22
21... 43 33
22... 54 35
23... 55 49
24... 69 53
25... 70 72
26... 87 78
27... 88 103 1
28... 109 112 1
29... 110 145 1
30... 136 160
31... 137 200 3
32... 168 220 3
MATHEMATICA
p[n_] := p[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
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]]}]];
c1[n_, k_] := c1[n, k] = Count[u[n], k];
m[n_] := -1 + Min[Flatten[Position[Table[c1[n, k], {k, 1, n + 1}], 0]]]
u = Table[c1[n, k], {n, 1, 50}, {k, 1, m[n]}]
TableForm[u] (* A279033 array *)
Flatten[u] (* A279033 sequence *)
CROSSREFS
KEYWORD
nonn,easy,tabf
AUTHOR
Clark Kimberling, Dec 04 2016
STATUS
approved