cp's OEIS Frontend

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'.

%I A277231 #21 Oct 23 2023 10:32:08
%S A277231 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,
%T A277231 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,
%U A277231 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
%N 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'.
%C A277231 The row length of this irregular triangular array is A000009(n).
%C A277231 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.)
%C A277231 The number of parts m of these, also called strict or fermionic, partitions is from m = 1, 2, ..., A003056(n).
%C A277231 The row sums give [1, 1, 3, 2, 4, 6, 6, 7, 11, 14, 14, 19, 22, 28, 36, ...].
%C A277231 For details, references and examples see A277230.
%e A277231 The irregular triangle begins (brackets separate partitions with equal number of parts m = 1, 2, 3, ..., A003056(n)):
%e A277231 n\k   1   2  3   4   5   6  7  8  9   10 ...
%e A277231 1:   [1]
%e A277231 2:   [1]
%e A277231 3:   [1] [2]
%e A277231 4:   [1] [1]
%e A277231 5:   [1] [1, 2]
%e A277231 6:   [1] [1, 1] [3]
%e A277231 7:   [1] [1, 1,  2] [1]
%e A277231 8:   [1] [1, 1,  1] [1,  2]
%e A277231 9:   [1] [1, 1,  1,  2] [1, 1, 3]
%e A277231 10:  [1] [1, 1,  1,  1] [1, 1, 2, 1] [4]
%e A277231 ...
%e A277231 n = 11: [1] [1, 1, 1, 1, 2] [1, 1, 1, 1, 2] [1],
%e A277231 n = 12: [1] [1, 1, 1, 1, 1] [1, 1, 1, 2, 1, 1, 3] [1, 2],
%e A277231 n = 13: [1] [1, 1, 1, 1, 1, 2] [1, 1, 1, 1, 1, 1, 2, 1] [1, 1, 3],
%e A277231 n = 14: [1] [1, 1, 1, 1, 1, 1] [1, 1, 1, 1, 2, 1, 1, 1, 1, 2] [1, 1, 2, 1, 4],
%e A277231 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].
%t A277231 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 *)
%Y A277231 Cf. A000009, A003056, A277230, A366509 (row records).
%K A277231 nonn,tabf,easy
%O A277231 1,4
%A A277231 _Wolfdieter Lang_, Oct 21 2016