A277231 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'.
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
Offset: 1
Examples
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].
Programs
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Mathematica
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 *)
Comments