cp's OEIS Frontend

A115621 Signature of partitions in Abramowitz and Stegun order.

%I A115621 #45 Jun 13 2020 00:49:18
%S A115621 1,1,2,1,1,1,3,1,1,1,2,1,2,4,1,1,1,1,1,1,2,1,2,1,3,5,1,1,1,1,1,2,1,2,
%T A115621 1,1,1,3,1,3,2,2,1,4,6,1,1,1,1,1,1,1,1,2,1,1,1,1,2,1,2,1,3,1,1,2,1,3,
%U A115621 1,4,2,3,1,5,7,1,1,1,1,1,1,1,2,1,2,1,1,1,1,1,1,1,2,1,2,1,3,1,1,2,2,2,1,1,2
%N A115621 Signature of partitions in Abramowitz and Stegun order.
%C A115621 The signature of a multiset is a partition consisting of the repetition factors of the original partition. Regarding a partition as a multiset, the signature of a partition is defined. E.g., [1,1,3,4,4] = [1^2,3^1,4^2], so the repetition factors are 2,1,2, making the signature [1,2,2] = [1,2^2]. Partitions are written here in increasing part size, so [1,2^2] is 1,2,2, not 2,2,1. - Edited by _Franklin T. Adams-Watters_, Jul 09 2012
%C A115621 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 A115621 Robert Price, <a href="/A115621/b115621.txt">Table of n, a(n) for n = 1..8266</a> (first 20 rows).
%H A115621 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%e A115621 [1];
%e A115621 [1], [2];
%e A115621 [1], [1,1], [3];
%e A115621 [1], [1,1], [2], [1,2], [4];
%e A115621 ...
%e A115621 From _Hartmut F. W. Hoft_, Apr 25 2015: (Start)
%e A115621 Extending the triangle to rows 5 and 6 where row headings indicate the number of elements in the underlying partitions. Brackets group the multiplicities of a single partition.
%e A115621     row 5         row 6
%e A115621 1:  [1]           [1]
%e A115621 2:  [1,1] [1,1]   [1,1] [1,1] [2]
%e A115621 3:  [1,2] [1,2]   [1,2] [1,1,1] [3]
%e A115621 4:  [1,3]         [1,3] [2,2]
%e A115621 5:  [5]           [1,4]
%e A115621 6:                [6]
%e A115621 (End)
%t A115621 (* row[] and triangle[] compute structured rows of the triangle as laid out above *)
%t A115621 mL[pL_] := Map[Last[Transpose[Tally[#]]]&, pL]
%t A115621 row[n_] := Map[Map[Sort, mL[#]]&, GatherBy[Map[Sort, IntegerPartitions[n]], Length]]
%t A115621 triangle[n_] := Map[row, Range[n]]
%t A115621 a115621[n_]:= Flatten[triangle[n]]
%t A115621 Take[a115621[8],105] (* data *)  (* _Hartmut F. W. Hoft_, Apr 25 2015 *)
%t A115621 Map[Sort[#, Less] &, Table[Last /@ Transpose /@ Tally /@ Sort[Reverse /@ IntegerPartitions[n]], {n, 8}], 2]
%o A115621 (SageMath) from collections import Counter
%o A115621 def A115621_row(n):
%o A115621     h = lambda p: sorted(Counter(p).values())
%o A115621     return flatten([h(p) for k in (0..n) for p in Partitions(n, length=k)])
%o A115621 for n in (1..10): print(A115621_row(n)) # _Peter Luschny_, Nov 02 2019
%Y A115621 Cf. A036036, A113787, A115622, A103921 (part counts), A000070 (row counts).
%K A115621 nonn,tabf
%O A115621 1,3
%A A115621 _Franklin T. Adams-Watters_, Jan 25 2006