OFFSET
1,3
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.
LINKS
Robert Price, Table of n, a(n) for n = 1..8266 (first 20 rows).
EXAMPLE
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)
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 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Franklin T. Adams-Watters, Jan 25 2006
STATUS
approved