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A366509 a(n) is the maximum number of dots on the slope of a Ferrers diagram of a partition of n into distinct parts.

%I A366509 #33 Mar 28 2025 03:45:08
%S A366509 1,1,2,1,2,3,2,2,3,4,2,3,3,4,5,3,3,4,4,5,6,4,4,4,5,5,6,7,4,5,5,5,6,6,
%T A366509 7,8,5,5,6,6,6,7,7,8,9,6,6,6,7,7,7,8,8,9,10,7,7,7,7,8,8,8,9,9,10,11,7,
%U A366509 8,8,8,8,9,9,9,10,10,11,12,8,8,9,9,9,9,10
%N A366509 a(n) is the maximum number of dots on the slope of a Ferrers diagram of a partition of n into distinct parts.
%C A366509 A Ferrers diagram arranges the parts of a partition in left-justified rows of dots, where the numbers of dots in row m corresponds to the m-th part of the partition, with parts in decreasing order.
%C A366509 The slope of a Ferrers diagram is the longest 45-degree line segment joining the rightmost dot in the first row with other dots in the diagram (see example).
%C A366509 If the top row of a diagram for n has A123578(n) dots, the corresponding slope is maximal.
%H A366509 Paolo Xausa, <a href="/A366509/b366509.txt">Table of n, a(n) for n = 1..10000</a>
%H A366509 Tom M. Apostol, <a href="https://doi.org/10.1007/978-1-4757-5579-4">Introduction to Analytic Number Theory</a>, Springer, New York, NY, 1976, pp. 313-315.
%H A366509 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FerrersDiagram.html">Ferrers Diagram</a>.
%F A366509 a(n) = r - A123578(A000217(r)-n), where r = A123578(n).
%F A366509 In particular, if n is a triangular number, a(n) = r.
%e A366509 The Ferrers diagrams for the partitions of n = 7 into distinct parts are:
%e A366509 .
%e A366509 .  (7)             (6,1)         (5,2)       (4,3)     (4,2,1)
%e A366509 .  o o o o o o o   o o o o o o   o o o o o   o o o o   o o o o
%e A366509 .                  o             o o         o o o     o o
%e A366509 .                                                      o
%e A366509 .
%e A366509 The maximal slope (joining 2 dots) corresponds to the (4,3) partition.
%e A366509 For n = 11 there are two diagrams with maximal slope (joining 2 dots):
%e A366509 .
%e A366509 .  o o o o o o   o o o o o
%e A366509 .  o o o o o     o o o o
%e A366509 .                o o
%e A366509 .
%e A366509 For n = 26 the maximal slope, corresponding to the partition (7,6,5,4,3,1), joins 5 dots:
%e A366509 .
%e A366509 .  o o o o o o o
%e A366509 .             /
%e A366509 .  o o o o o o
%e A366509 .           /
%e A366509 .  o o o o o
%e A366509 .         /
%e A366509 .  o o o o
%e A366509 .       /
%e A366509 .  o o o
%e A366509 .
%e A366509 .  o
%e A366509 .
%t A366509 A123578[n_]:=Floor[Sqrt[2n]+1/2];
%t A366509 A366509[n_]:=With[{r=A123578[n]},r-A123578[PolygonalNumber[r]-n]];
%t A366509 Array[A366509,100]
%Y A366509 Cf. A000009, A000217, A123578.
%Y A366509 Row records in A277231.
%K A366509 nonn
%O A366509 1,3
%A A366509 _Paolo Xausa_, Oct 11 2023