OFFSET
1,2
COMMENTS
Let m and n be two positive integers such that m <= n. It appears that any set formed by m connected sections, or m disconnected sections, or a mixture of both, has the same properties described in the section example. (Cf. A135010, A207031, A207032, A212010). - Omar E. Pol, May 01 2012
FORMULA
EXAMPLE
Calculation for n = 6. Write the partitions of 6 and below the sums of their columns:
.
. 6
. 3 + 3
. 4 + 2
. 2 + 2 + 2
. 5 + 1
. 3 + 2 + 1
. 4 + 1 + 1
. 2 + 2 + 1 + 1
. 3 + 1 + 1 + 1
. 2 + 1 + 1 + 1 + 1
. 1 + 1 + 1 + 1 + 1 + 1
. ------------------------
. 35, 16, 8, 4, 2, 1 --> Row 6 of triangle A181187.
. | /| /| /| /| /|
. | / | / | / | / | / |
. |/ |/ |/ |/ |/ |
. 19, 8, 4, 2, 1, 1 --> Row 6 of triangle A066633.
.
More generally, it appears that the sum of column k is also the total number of parts >= k in all partitions of n. It appears that the first differences of the column sums together with 1 give the number of occurrences of k in all partitions of n.
On the other hand we can see that the partitions of 6 contain:
24 odd parts >= 1 (the odd parts).
11 even parts >= 2 (the even parts).
5 odd parts >= 3.
3 even parts >= 4.
2 odd parts >= 5.
1 even part >= 6.
Then, using the values of the column sums, it appears that:
T(6,1) = 35 - 16 + 8 - 4 + 2 - 1 = 24
T(6,2) = 16 - 8 + 4 - 2 + 1 = 11
T(6,3) = 8 - 4 + 2 - 1 = 5
T(6,4) = 4 - 2 + 1 = 3
T(6,5) = 2 - 1 = 1
T(6,6) = 1 = 1
So the 6th row of our triangle gives 24, 11, 5, 3, 1, 1.
Finally, for all partitions of 6, we can write:
The number of odd parts is equal to T(6,1) = 24.
The number of even parts is equal to T(6,2) = 11.
The number of odd parts >= 3 is equal to T(6,3) = 5.
The number of even parts >= 4 is equal to T(6,4) = 3.
The number of odd parts >= 5 is equal to T(6,5) = 1.
The number of even parts >= 6 is equal to T(6,6) = 1.
More generally, we can write the same properties for any positive integer.
Triangle begins:
1;
2, 1;
5, 1, 1;
8, 4, 1, 1;
15, 5, 3, 1, 1;
24, 11, 5, 3, 1, 1;
39, 15, 9, 4, 3, 1, 1;
58, 28, 13, 9, 4, 3, 1, 1;
90, 38, 23, 12, 8, 4, 3, 1, 1;
130, 62, 33, 21, 12, 8, 4, 3, 1, 1;
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Feb 15 2012
EXTENSIONS
More terms from Alois P. Heinz, Feb 18 2012
STATUS
approved