|
| |
|
|
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.
|
|
1
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
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).
|
|
|
REFERENCES
|
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.
|
|
|
LINKS
|
|
|
|
FORMULA
|
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.
|
|
|
EXAMPLE
|
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].
|
|
|
MATHEMATICA
|
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 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,tabf,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
