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

1,3

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.

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.

Formula

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

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

Example

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
.

Mathematica

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

Crossrefs

Cf. A000009, A000217, A123578.

Row records in A277231.

Sequence in context: A284566 A079056 A341839 * A231205 A003984 A087061

Adjacent sequences: A366506 A366507 A366508 * A366510 A366511 A366512

Keyword

nonn

Author

Paolo Xausa, Oct 11 2023

Status

approved