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A060857 Describe all the numbers already used (sorted into increasing order - not splitting numbers up into their digits).

%I A060857 #18 Jan 11 2021 18:14:39
%S A060857 1,11,31,4113,612314,8112332416,1113253342618,131528344153628111,
%T A060857 1617210364354648211113,181921239445661758110311213116,
%U A060857 2211121431146586276829210411112313216118
%N A060857 Describe all the numbers already used (sorted into increasing order - not splitting numbers up into their digits).
%H A060857 Reinhard Zumkeller, <a href="/A060857/b060857.txt">Table of n, a(n) for n = 0..60</a>
%H A060857 Onno M. Cain and Sela T. Enin, <a href="https://arxiv.org/abs/2004.00209">Inventory Loops (i.e. Counting Sequences) have Pre-period 2 max S_1 + 60</a>, arXiv:2004.00209 [math.NT], 2020.
%e A060857 One; one one; three ones; four ones, one three; six ones, two threes, one four; eight ones, one two, three threes, two fours, one six; eleven ones, three twos, five threes, three fours, two sixes, one eight; thirteen [note not 15] ones, five twos, eight threes, four fours, one five, three sixes, two eights, one eleven [note than numbers >9 are preserved as wholes rather than as a collection of digits]; etc.
%t A060857 FromDigits /@ Nest[Append[#, Flatten@ Map[Reverse, Tally@ Sort@ Flatten@ # ] ] &, {{1}}, 10] (* _Michael De Vlieger_, Jul 15 2020 *)
%o A060857 (Haskell)
%o A060857 import Data.List (group, sort, transpose)
%o A060857 a060857 n = a060857_list !! n
%o A060857 a060857_list = 1 : f [1] :: [Integer] where
%o A060857    f xs = (read $ concatMap show ys) : f (xs ++ ys) where
%o A060857           ys = concat $ transpose [map length zss, map head zss]
%o A060857           zss = group $ sort xs
%o A060857 -- _Reinhard Zumkeller_, Jan 25 2014
%o A060857 (Python)
%o A060857 def summarize_lst(lst):
%o A060857   ans = []
%o A060857   for d in sorted(set(lst)): ans += [lst.count(d), d]
%o A060857   return ans
%o A060857 def aupton(nn):
%o A060857   alst, arunninglst = [1], [1]
%o A060857   for n in range(nn):
%o A060857     nxt_lst = summarize_lst(arunninglst)
%o A060857     arunninglst += nxt_lst
%o A060857     alst.append(int("".join(map(str, nxt_lst))))
%o A060857   return alst
%o A060857 print(aupton(10)) # _Michael S. Branicky_, Jan 11 2021
%Y A060857 This is a combination of methods used in A005151 and A045982. The first word of each term (the number of ones used earlier) seems to be equal to A030711 and A030761.
%K A060857 base,nice,nonn
%O A060857 0,2
%A A060857 _Henry Bottomley_, May 03 2001