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a(n) is the maximum number of dots on the slope of a Ferrers diagram of a partition of n into distinct parts.
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%I #30 Oct 20 2023 06:47:31

%S 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 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 8,8,8,8,9,9,9,10,10,11,12,8,8,9,9,9,9,10

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

%C 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 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 If the top row of a diagram for n has A123578(n) dots, the corresponding slope is maximal.

%H Paolo Xausa, <a href="/A366509/b366509.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FerrersDiagram.html">Ferrers Diagram</a>.

%F a(n) = r - A123578(A000217(r)-n)), where r = A123578(n).

%F In particular, if n is a triangular number, a(n) = r.

%e The Ferrers diagrams for the partitions of n = 7 into distinct parts are:

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%e . (7) (6,1) (5,2) (4,3) (4,2,1)

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%e The maximal slope (joining 2 dots) corresponds to the (4,3) partition.

%e For n = 11 there are two diagrams with maximal slope (joining 2 dots):

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%e For n = 26 the maximal slope, corresponding to the partition (7,6,5,4,3,1), joins 5 dots:

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%t A123578[n_]:=Floor[Sqrt[2n]+1/2];

%t A366509[n_]:=With[{r=A123578[n]},r-A123578[PolygonalNumber[r]-n]];

%t Array[A366509,100]

%Y Cf. A000009, A000217, A123578.

%Y Row records in A277231.

%K nonn

%O 1,3

%A _Paolo Xausa_, Oct 11 2023