A277231 -id:A277231 - cp's OEIS frontend

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

Paolo Xausa, Oct 11 2023

nonn

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.

			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
.

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.

Cf. A000009, A000217, A123578.

Row records in A277231.

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

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

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

A277230 Irregular triangular array T(n, k) giving in row n the base 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. T(n, k) is the smallest part of the k-th partition of n into distinct parts.

Original entry on oeis.org

1, 2, 3, 1, 4, 1, 5, 1, 2, 6, 1, 2, 1, 7, 1, 2, 3, 1, 8, 1, 2, 3, 1, 1, 9, 1, 2, 3, 4, 1, 1, 2, 10, 1, 2, 3, 4, 1, 1, 1, 2, 1, 11, 1, 2, 3, 4, 5, 1, 1, 1, 2, 2, 1, 12, 1, 2, 3, 4, 5, 1, 1, 1, 1, 2, 2, 3, 1, 1, 13, 1, 2, 3, 4, 5, 6, 1, 1, 1, 1, 2, 2, 2, 3, 1, 1, 1, 14, 1, 2, 3, 4, 5, 6, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 1, 1, 1, 1, 2, 15, 1, 2, 3, 4, 5, 6, 7, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Wolfdieter Lang, Oct 21 2016

The length of row n of this irregular triangular array is A000009(n).
For the Abramowitz-Stegun order of partitions see an Apr 04 2011 comment on A036036.
The sum of the numbers of row n is A092265(n).
See the Hardy (H) and Hardy-Wright (H-W) references, where the base is called beta. The companion array is A277231 giving the slopes (called sigma) of these partitions with distinct parts. These beta and sigma numbers play a role in an elementary proof of Euler's pentagonal-number theorem (pp. 284-287 in (H-W), and pp. 83-85 in (H)) by F. Franklin from 1881. See also MacMahon and Charalambides.
The base of the Ferrers diagram of the k-th partition of n into distinct parts (in the mentioned order) is the number of nodes in the last row, the smallest part of the partition.
The slope of such a partition is the number of nodes on the NE-SW diagonal through the last node of the first row of the Ferrers diagram. (The name may be misleading. The usual slope of the NE-SW diagonal is of course 1).
The number of parts m of these partitions is from m = 1, 2, ..., A003056(n).

			The irregular triangle begins (brackets separate partitions with equal number of parts m = 1, 2, 3,..., A003056(n)):
n\k   1   2  3   4   5   6   7  8  9  10 ...
1:   [1]
2:   [2]
3:   [3] [1]
4:   [4] [1]
5:   [5] [1, 2]
6:   [6] [1, 2] [1]
7:   [7] [1, 2,  3] [1]
8:   [8] [1, 2,  3] [1,  1]
9:   [9] [1, 2,  3,  4] [1,  1, 2]
10: [10] [1, 2,  3,  4] [1,  1, 1, 2] [1]
...
n = 11: [11] [1, 2,  3,  4, 5] [1, 1, 1, 2, 2] [1],
n = 12: [12] [1, 2, 3, 4, 5] [1, 1, 1, 1, 2, 2, 3] [1, 1],
n = 13: [13] [1, 2, 3, 4, 5, 6] [1, 1, 1, 1, 2, 2, 2, 3] [1, 1, 1],
n = 14: [14] [1, 2, 3, 4, 5, 6] [1, 1, 1, 1, 1, 2, 2, 2, 3, 3] [1, 1, 1, 1, 2],
n = 15: [15] [1, 2, 3, 4, 5, 6, 7] [1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4] [1, 1, 1, 1, 1, 2] [1].
----------------------------------------
The partition of n = 5 + 4 + 1 = 10 has base 1 and slope 2 (beta < sigma):
o o o o o
o o o o
o
The partition of n = 5 + 3 + 1 = 9 has base 1 and slope 1 (beta = sigma):
o o o o o
o o o
o
The partition of n = 5 + 3 + 2 = 10 has base 2 and slope 1 (beta > sigma):
o o o o o
o o o
o o
------------------------------------------
The partitions of n = 6 with m = 1, 2, and 3,  (3 = A003056(6)) distinct parts are:
[6], [[5, 1], [4, 2]], [3, 2, 1], with base numbers in row n=6: [6] [1, 2] [1]
and slope numbers in row n=6 of A277231:
[1] [1, 1] [3].

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, pp. 389-391, 396, 595.
G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 83-85.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Clarendon Press, Oxford, 2003, pp. 284-287.
P. A. MacMahon, Combinatory Analysis, Vol. II, Chelsea Publishing Company, New York, 1960, pp. 21-23.

F. Franklin, Sur le développement du produit infini (1-x) (1-x^2) (1-x^3) ..., Comptes Rendus de l'Académie des Sciences, Paris, 92 (1881) 448-450.

Cf. A000009, A003056, A092265, A277231.

Table[Function[w, Flatten@ Map[Function[k, Min /@ 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 *)

T(n, k) is the smallest part of the k-th partition of n into distinct parts. n >=1. k=1, 2, ..., A000009(n). Partitions appear in Abramowitz-Stegun order.

cp's OEIS Frontend

A366509 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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