|
%I #25 Sep 19 2015 11:53:55
%S 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 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 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 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 The number of t's in row n gives A097364(n,t), with n>=1 and 0<=t<n.
%C Rows converge to A244967, which is A141285 - 1.
%C Row n has length A000041(n).
%C Row sums give A116686.
%H 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 T(n,k) = A141285(k) - A196931(n,k), n>=1, 1<=k<=A000041(n).
%e Triangle begins:
%e 0;
%e 0, 0;
%e 0, 1, 0;
%e 0, 1, 2, 0, 0;
%e 0, 1, 2, 1, 3, 1, 0;
%e 0, 1, 2, 1, 3, 2, 4, 0, 2, 0, 0;
%e 0, 1, 2, 1, 3, 2, 4, 1, 3, 2, 5, 1, 3, 1, 0;
%e 0, 1, 2, 1, 3, 2, 4, 1, 3, 2, 5, 2, 4, 3, 6, 0, 2, 1, 4, 2, 0, 0;
%e ...
%e For n = 6 we have:
%e --------------------------------------------------------
%e . Largest Smallest Difference
%e k Partition of 6 part part T(6,k)
%e --------------------------------------------------------
%e 1: [1, 1, 1, 1, 1, 1] 1 - 1 = 0
%e 2: [2, 1, 1, 1, 1] 2 - 1 = 1
%e 3: [3, 1, 1, 1] 3 - 1 = 2
%e 4: [2, 2, 1, 1] 2 - 1 = 1
%e 5: [4, 1, 1] 4 - 1 = 3
%e 6: [3, 2, 1] 3 - 1 = 2
%e 7: [5, 1] 5 - 1 = 4
%e 8: [2, 2, 2] 2 - 2 = 0
%e 9: [4, 2] 4 - 2 = 2
%e 10: [3, 3] 3 - 3 = 0
%e 11: [6] 6 - 6 = 0
%e --------------------------------------------------------
%e 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 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 Cf. A000005, A000041, A008805, A049820, A097364, A116686, A128508, A135010, A141285, A196931, A218567-A218573, A244967.
%K nonn,tabf
%O 1,9
%A _Omar E. Pol_, Jul 18 2014
|
