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

%I A115622 #27 Jun 14 2020 12:55:53
%S A115622 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 A115622 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 A115622 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 A115622 Irregular triangle read by rows: row m lists the signatures of all partitions of m when the partitions are arranged in Mathematica order.
%C A115622 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 A115622 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 A115622 Robert Price, <a href="/A115622/b115622.txt">Table of n, a(n) for n = 1..8266</a> (first 20 rows).
%e A115622 From _Hartmut F. W. Hoft_, Apr 25 2015: (Start)
%e A115622 The first six rows of the triangle are as follows.
%e A115622 1:  [1]
%e A115622 2:  [1] [2]
%e A115622 3:  [1] [1,1] [3]
%e A115622 4:  [1] [1,1] [2]   [2,1] [4]
%e A115622 5:  [1] [1,1] [1,1] [2,1] [2,1] [3,1]   [5]
%e A115622 6:  [1] [1,1] [1,1] [2,1] [2]   [1,1,1] [3,1] [3] [2,2] [4,1] [6]
%e A115622 See A115621 for the signatures in Abramowitz-Stegun order.
%e A115622 (End)
%t A115622 (* row[] and triangle[] compute structured rows of the triangle as laid out above *)
%t A115622 mL[pL_] := Map[Last[Transpose[Tally[#]]]&, pL]
%t A115622 row[n_] := Map[Reverse[Sort[#]]&, mL[IntegerPartitions[n]]]
%t A115622 triangle[n_] := Map[row, Range[n]]
%t A115622 a115622[n_]:= Flatten[triangle[n]]
%t A115622 Take[a115622[8],105] (* data *)  (* _Hartmut F. W. Hoft_, Apr 25 2015 *)
%t A115622 Map[Sort[#, Greater] &, Table[Last /@ Transpose /@ Tally /@ IntegerPartitions[n], {n, 8}], 2] // Flatten  (* _Robert Price_, Jun 12 2020 *)
%Y A115622 Cf. A080577, A115624, A115621, part counts A115623, row counts A000070.
%K A115622 nonn,tabf
%O A115622 1,3
%A A115622 _Franklin T. Adams-Watters_, Jan 25 2006