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A244966 Triangle read by rows: T(n,k) is the difference between the largest and the smallest part of the k-th partition in the list of colexicographically ordered partitions of n, with n>=1 and 1<=k<=p(n), where p(n) is the number of partitions of n. 2
0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 1, 3, 1, 0, 0, 1, 2, 1, 3, 2, 4, 0, 2, 0, 0, 0, 1, 2, 1, 3, 2, 4, 1, 3, 2, 5, 1, 3, 1, 0, 0, 1, 2, 1, 3, 2, 4, 1, 3, 2, 5, 2, 4, 3, 6, 0, 2, 1, 4, 2, 0, 0, 0, 1, 2, 1, 3, 2, 4, 1, 3, 2, 5, 2, 4, 3, 6, 1, 3, 2, 5, 4, 3, 7, 1, 3, 2, 5, 0, 3, 1, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,9

COMMENTS

The number of t's in row n gives A097364(n,t), with n>=1 and 0<=t<n.

Rows converge to A244967, which is A141285 - 1.

Row n has length A000041(n).

Row sums give A116686.

LINKS

Table of n, a(n) for n=1..96.

G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014

FORMULA

T(n,k) = A141285(k) - A196931(n,k), n>=1, 1<=k<=A000041(n).

EXAMPLE

Triangle begins:

0;

0, 0;

0, 1, 0;

0, 1, 2, 0, 0;

0, 1, 2, 1, 3, 1, 0;

0, 1, 2, 1, 3, 2, 4, 0, 2, 0, 0;

0, 1, 2, 1, 3, 2, 4, 1, 3, 2, 5, 1, 3, 1, 0;

0, 1, 2, 1, 3, 2, 4, 1, 3, 2, 5, 2, 4, 3, 6, 0, 2, 1, 4, 2, 0, 0;

...

For n = 6 we have:

--------------------------------------------------------

.                        Largest  Smallest   Difference

k    Partition of 6        part     part       T(6,k)

--------------------------------------------------------

1:  [1, 1, 1, 1, 1, 1]      1    -    1     =     0

2:  [2, 1, 1, 1, 1]         2    -    1     =     1

3:  [3, 1, 1, 1]            3    -    1     =     2

4:  [2, 2, 1, 1]            2    -    1     =     1

5:  [4, 1, 1]               4    -    1     =     3

6:  [3, 2, 1]               3    -    1     =     2

7:  [5, 1]                  5    -    1     =     4

8:  [2, 2, 2]               2    -    2     =     0

9:  [4, 2]                  4    -    2     =     2

10: [3, 3]                  3    -    3     =     0

11: [6]                     6    -    6     =     0

--------------------------------------------------------

So the 6th row of triangle is [0,1,2,1,3,2,4,0,2,0,0] and the row sum is A116686(6) = 15.

Note that in the 6th row there are four 0's so A097364(6,0) = 4, there are two 1's so A097364(6,1) = 2, there are three 2's so A097364(6,2) = 3, there is only one 3 so A097364(6,3) = 1, there is only one 4 so A097364(6,4) = 1 and there are no 5's so A097364(6,5) = 0.

CROSSREFS

Cf. A000005, A000041, A008805,  A049820, A097364, A116686, A128508, A135010, A141285, A196931, A218567-A218573, A244967.

Sequence in context: A228360 A303138 A276205 * A079100 A296167 A244233

Adjacent sequences:  A244963 A244964 A244965 * A244967 A244968 A244969

KEYWORD

nonn,tabf

AUTHOR

Omar E. Pol, Jul 18 2014

STATUS

approved

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Last modified March 26 18:36 EDT 2019. Contains 321511 sequences. (Running on oeis4.)