A244967 -id:A244967 - cp's OEIS frontend

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.

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

Offset: 1

Omar E. Pol, Jul 18 2014

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

Rows converge to A244967, which is A141285 - 1.

Row n has length A000041(n).

Row sums give A116686.

			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.

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

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

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

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cp's OEIS Frontend

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.

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