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A279033 Irregular triangular array: T(n,i) = number of strict partitions of n having crossover index k; see Comments.

%I A279033 #8 Dec 07 2016 10:20:34
%S A279033 1,1,2,2,3,4,5,6,7,1,9,1,10,2,13,2,14,4,18,4,19,8,24,8,25,13,32,14,33,
%T A279033 21,42,22,43,33,54,35,55,49,69,53,70,72,87,78,88,103,1,109,112,1,110,
%U A279033 145,1,136,160,137,200,3,168,220,2,169,275,4,206,303,3
%N A279033 Irregular triangular array: T(n,i) = number of strict partitions of n having crossover index k; see Comments.
%C A279033 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.
%C A279033 First 32 rows (indexed by column 1):
%C A279033 1...     1
%C A279033 2...     1
%C A279033 3...     2
%C A279033 4...     2
%C A279033 5...     3
%C A279033 6...     4
%C A279033 7...     5
%C A279033 8...     6
%C A279033 9...     7      1
%C A279033 10...    9      1
%C A279033 11...    10     2
%C A279033 12...    13     2
%C A279033 13...    14     4
%C A279033 14...    18     4
%C A279033 15...    19     8
%C A279033 16...    24     8
%C A279033 17...    25     13
%C A279033 18...    32     14
%C A279033 19...    33     21
%C A279033 20...    42     22
%C A279033 21...    43     33
%C A279033 22...    54     35
%C A279033 23...    55     49
%C A279033 24...    69     53
%C A279033 25...    70     72
%C A279033 26...    87     78
%C A279033 27...    88    103    1
%C A279033 28...    109   112    1
%C A279033 29...    110   145    1
%C A279033 30...    136   160
%C A279033 31...    137   200    3
%C A279033 32...    168   220    3
%t A279033 p[n_] := p[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
%t A279033 t[n_, k_] := t[n, k] = p[n][[k]];
%t A279033 q[n_, k_] := q[n, k] = Select[Range[50], Sum[t[n, k][[i]], {i, 1, #}] >= n/2 &, 1];
%t A279033 u[n_] := u[n] = Flatten[Table[q[n, k], {k, 1, Length[p[n]]}]];
%t A279033 c1[n_, k_] := c1[n, k] = Count[u[n], k];
%t A279033 m[n_] := -1 + Min[Flatten[Position[Table[c1[n, k], {k, 1, n + 1}], 0]]]
%t A279033 u = Table[c1[n, k], {n, 1, 50}, {k, 1, m[n]}]
%t A279033 TableForm[u] (* A279033 array *)
%t A279033 Flatten[u]   (* A279033 sequence *)
%Y A279033 Cf. A000009, A088567, A276468.
%K A279033 nonn,easy,tabf
%O A279033 1,3
%A A279033 _Clark Kimberling_, Dec 04 2016