A115622 Irregular triangle read by rows: row m lists the signatures of all partitions of m when the partitions are arranged in Mathematica order.

1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 4, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 5, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 3, 2, 2, 4, 1, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 1, 4, 1, 3, 1, 3, 2, 5, 1, 7, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 4, 1, 2, 1, 2, 2, 2

Offset: 1

Author

Franklin T. Adams-Watters, Jan 25 2006

Keywords

• nonn
• tabf

Comments

The signature of a partition is a partition consisting of the repetition factors of the original partition. E.g., [4,4,3,1,1] = [4^2,3^1,1^2], so the repetition factors are 2,1,2, making the signature [2,2,1] = [2^2,1].

The sum (or order) of the signature is the number of parts of the original partition and the number of parts of the signature is the number of distinct parts of the original partition.

Examples

			From _Hartmut F. W. Hoft_, Apr 25 2015: (Start)
The first six rows of the triangle are as follows.
1:  [1]
2:  [1] [2]
3:  [1] [1,1] [3]
4:  [1] [1,1] [2]   [2,1] [4]
5:  [1] [1,1] [1,1] [2,1] [2,1] [3,1]   [5]
6:  [1] [1,1] [1,1] [2,1] [2]   [1,1,1] [3,1] [3] [2,2] [4,1] [6]
See A115621 for the signatures in Abramowitz-Stegun order.
(End)
		

Links

• Robert Price, Table of n, a(n) for n = 1..8266 (first 20 rows).

Crossrefs

Cf. A080577, A115624, A115621, part counts A115623, row counts A000070.

Programs

Mathematica

(* row[] and triangle[] compute structured rows of the triangle as laid out above *)
mL[pL_] := Map[Last[Transpose[Tally[#]]]&, pL]
row[n_] := Map[Reverse[Sort[#]]&, mL[IntegerPartitions[n]]]
triangle[n_] := Map[row, Range[n]]
a115622[n_]:= Flatten[triangle[n]]
Take[a115622[8],105] (* data *)  (* Hartmut F. W. Hoft, Apr 25 2015 *)
Map[Sort[#, Greater] &, Table[Last /@ Transpose /@ Tally /@ IntegerPartitions[n], {n, 8}], 2] // Flatten  (* Robert Price, Jun 12 2020 *)