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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.

%I A244966 #25 Sep 19 2015 11:53:55
%S A244966 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,
%T A244966 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,
%U A244966 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
%N 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.
%C A244966 The number of t's in row n gives A097364(n,t), with n>=1 and 0<=t<n.
%C A244966 Rows converge to A244967, which is A141285 - 1.
%C A244966 Row n has length A000041(n).
%C A244966 Row sums give A116686.
%H A244966 G. E. Andrews, M. Beck and N. Robbins, <a href="http://arxiv.org/abs/1406.3374">Partitions with fixed differences between largest and smallest parts</a>, arXiv:1406.3374 [math.NT], 2014
%F A244966 T(n,k) = A141285(k) - A196931(n,k), n>=1, 1<=k<=A000041(n).
%e A244966 Triangle begins:
%e A244966 0;
%e A244966 0, 0;
%e A244966 0, 1, 0;
%e A244966 0, 1, 2, 0, 0;
%e A244966 0, 1, 2, 1, 3, 1, 0;
%e A244966 0, 1, 2, 1, 3, 2, 4, 0, 2, 0, 0;
%e A244966 0, 1, 2, 1, 3, 2, 4, 1, 3, 2, 5, 1, 3, 1, 0;
%e A244966 0, 1, 2, 1, 3, 2, 4, 1, 3, 2, 5, 2, 4, 3, 6, 0, 2, 1, 4, 2, 0, 0;
%e A244966 ...
%e A244966 For n = 6 we have:
%e A244966 --------------------------------------------------------
%e A244966 .                        Largest  Smallest   Difference
%e A244966 k    Partition of 6        part     part       T(6,k)
%e A244966 --------------------------------------------------------
%e A244966 1:  [1, 1, 1, 1, 1, 1]      1    -    1     =     0
%e A244966 2:  [2, 1, 1, 1, 1]         2    -    1     =     1
%e A244966 3:  [3, 1, 1, 1]            3    -    1     =     2
%e A244966 4:  [2, 2, 1, 1]            2    -    1     =     1
%e A244966 5:  [4, 1, 1]               4    -    1     =     3
%e A244966 6:  [3, 2, 1]               3    -    1     =     2
%e A244966 7:  [5, 1]                  5    -    1     =     4
%e A244966 8:  [2, 2, 2]               2    -    2     =     0
%e A244966 9:  [4, 2]                  4    -    2     =     2
%e A244966 10: [3, 3]                  3    -    3     =     0
%e A244966 11: [6]                     6    -    6     =     0
%e A244966 --------------------------------------------------------
%e A244966 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.
%e A244966 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.
%Y A244966 Cf. A000005, A000041, A008805,  A049820, A097364, A116686, A128508, A135010, A141285, A196931, A218567-A218573, A244967.
%K A244966 nonn,tabf
%O A244966 1,9
%A A244966 _Omar E. Pol_, Jul 18 2014