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A277231
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Irregular triangular array T(n, k) giving in row n the so-called slope of the Ferrers diagram of the k-th partition of n into distinct parts. The partitions of n are taken in Abramowitz-Stegun order but with decreasing parts. See a comment for the definition of this 'slope'.
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2
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1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 5
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OFFSET
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1,4
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COMMENTS
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The row length of this irregular triangular array is A000009(n).
The slope T(n, k) of the k-th partition of n into distinct parts is here defined as the number of nodes of the Ferrers diagram (rows with falling parts) that lie on the NE-SW diagonal through the last node on the first row. (This diagonal has, of course, the usual slope 1.)
The number of parts m of these, also called strict or fermionic, partitions is from m = 1, 2, ..., A003056(n).
The row sums give [1, 1, 3, 2, 4, 6, 6, 7, 11, 14, 14, 19, 22, 28, 36, ...].
For details, references and examples see A277230.
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LINKS
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EXAMPLE
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The irregular triangle begins (brackets separate partitions with equal number of parts m = 1, 2, 3, ..., A003056(n)):
n\k 1 2 3 4 5 6 7 8 9 10 ...
1: [1]
2: [1]
3: [1] [2]
4: [1] [1]
5: [1] [1, 2]
6: [1] [1, 1] [3]
7: [1] [1, 1, 2] [1]
8: [1] [1, 1, 1] [1, 2]
9: [1] [1, 1, 1, 2] [1, 1, 3]
10: [1] [1, 1, 1, 1] [1, 1, 2, 1] [4]
...
n = 11: [1] [1, 1, 1, 1, 2] [1, 1, 1, 1, 2] [1],
n = 12: [1] [1, 1, 1, 1, 1] [1, 1, 1, 2, 1, 1, 3] [1, 2],
n = 13: [1] [1, 1, 1, 1, 1, 2] [1, 1, 1, 1, 1, 1, 2, 1] [1, 1, 3],
n = 14: [1] [1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 2, 1, 1, 1, 1, 2] [1, 1, 2, 1, 4],
n = 15: [1] [1, 1, 1, 1, 1, 1, 2] [1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3] [1, 1, 1, 1, 2, 1] [5].
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MATHEMATICA
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Table[Function[w, Flatten@ Map[Function[k, 1 + Count[ TakeWhile[ Abs@ Differences@ #, # == 1 &], 1] & /@ Select[w, Length@ # == k &]], Range@ Max@ Map[Length, w]]]@ Select[DeleteCases[IntegerPartitions@ n, w_ /; MemberQ[Differences@ w, 0]], Length@ # <= Floor[(Sqrt[1 + 8 n] - 1)/2] &], {n, 15}] // Flatten (* Michael De Vlieger, Oct 26 2016 *)
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CROSSREFS
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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STATUS
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approved
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