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

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, 21, 42, 22, 43, 33, 54, 35, 55, 49, 69, 53, 70, 72, 87, 78, 88, 103, 1, 109, 112, 1, 110, 145, 1, 136, 160, 137, 200, 3, 168, 220, 2, 169, 275, 4, 206, 303, 3

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

Links

Table of n, a(n) for n=1..67.

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

Cf. A000009, A088567, A276468.

Sequence in context: A028391 A038668 A251629 * A304744 A327225 A071754

Adjacent sequences: A279030 A279031 A279032 * A279034 A279035 A279036

Keyword

nonn,easy,tabf

Author

Clark Kimberling, Dec 04 2016

Status

approved