A115622 - OEIS

%I #27 Jun 14 2020 12:55:53

%S 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,

%T 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,

%U 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

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

%C 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].

%C 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.

%H Robert Price, <a href="/A115622/b115622.txt">Table of n, a(n) for n = 1..8266</a> (first 20 rows).

%e From _Hartmut F. W. Hoft_, Apr 25 2015: (Start)

%e The first six rows of the triangle are as follows.

%e 1:  [1]

%e 2:  [1] [2]

%e 3:  [1] [1,1] [3]

%e 4:  [1] [1,1] [2]   [2,1] [4]

%e 5:  [1] [1,1] [1,1] [2,1] [2,1] [3,1]   [5]

%e 6:  [1] [1,1] [1,1] [2,1] [2]   [1,1,1] [3,1] [3] [2,2] [4,1] [6]

%e See A115621 for the signatures in Abramowitz-Stegun order.

%e (End)

%t (* row[] and triangle[] compute structured rows of the triangle as laid out above *)

%t mL[pL_] := Map[Last[Transpose[Tally[#]]]&, pL]

%t row[n_] := Map[Reverse[Sort[#]]&, mL[IntegerPartitions[n]]]

%t triangle[n_] := Map[row, Range[n]]

%t a115622[n_]:= Flatten[triangle[n]]

%t Take[a115622[8],105] (* data *)  (* _Hartmut F. W. Hoft_, Apr 25 2015 *)

%t Map[Sort[#, Greater] &, Table[Last /@ Transpose /@ Tally /@ IntegerPartitions[n], {n, 8}], 2] // Flatten  (* _Robert Price_, Jun 12 2020 *)

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

%K nonn,tabf

%O 1,3

%A _Franklin T. Adams-Watters_, Jan 25 2006