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A244966
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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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2
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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
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OFFSET
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1,9
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COMMENTS
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The number of t's in row n gives A097364(n,t), with n>=1 and 0<=t<n.
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LINKS
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FORMULA
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EXAMPLE
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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.
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CROSSREFS
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Cf. A000005, A000041, A008805, A049820, A097364, A116686, A128508, A135010, A141285, A196931, A218567-A218573, A244967.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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