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

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, 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, 8, 8, 8, 8, 9, 9, 9, 10, 10, 11, 12, 8, 8, 9, 9, 9, 9, 10

Offset: 1

Author

Paolo Xausa, Oct 11 2023

Keywords

• nonn

Comments

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.

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).

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

Examples

			The Ferrers diagrams for the partitions of n = 7 into distinct parts are:
.
.  (7)             (6,1)         (5,2)       (4,3)     (4,2,1)
.  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
.                  o             o o         o o o     o o
.                                                      o
.
The maximal slope (joining 2 dots) corresponds to the (4,3) partition.
For n = 11 there are two diagrams with maximal slope (joining 2 dots):
.
.  o o o o o o   o o o o o
.  o o o o o     o o o o
.                o o
.
For n = 26 the maximal slope, corresponding to the partition (7,6,5,4,3,1), joins 5 dots:
.
.  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
.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000
• Tom M. Apostol, Introduction to Analytic Number Theory, Springer, New York, NY, 1976, pp. 313-315.
• Eric Weisstein's World of Mathematics, Ferrers Diagram.

Crossrefs

Cf. A000009, A000217, A123578.

Row records in A277231.

Programs

Mathematica

A123578[n_]:=Floor[Sqrt[2n]+1/2];
A366509[n_]:=With[{r=A123578[n]},r-A123578[PolygonalNumber[r]-n]];
Array[A366509,100]

Formula

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

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