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%I #42 Jul 01 2023 09:28:27
%S 0,0,0,1,0,1,1,2,2,3,0,1,1,2,2,3,1,2,2,3,3,4,2,3,3,4,4,5,3,4,4,5,5,6,
%T 0,1,1,2,2,3,1,2,2,3,3,4,2,3,3,4,4,5,3,4,4,5,5,6,1,2,2,3,3,4,2,3,3,4,
%U 4,5,3,4,4,5,5,6,4,5,5,6,6,7,2,3,3,4,4,5,3,4,4,5,5,6,4,5,5,6,6,7,5,6,6,7,7
%N Array of digit sums of factorial representation of numbers 0,1,...,n!-1 for n >= 1.
%C The row lengths sequence is A000142 (factorials).
%C When the factorial representation is read as (D. N.) Lehmer code for permutations of n objects then the digit sums in row n count the inversions of the permutations arranged in lexicographic order.
%C Row n is the first n! terms of A034968. - _Franklin T. Adams-Watters_, May 13 2009
%H Alois P. Heinz, <a href="/A139365/b139365.txt">Rows n = 0..8, flattened</a>
%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/St000018">The number of inversions of a permutation </a>
%H A. Kohnert, <a href="ftp://ftp.mathe2.uni-bayreuth.de/axel/combalg/combalg.ps"> Kombinatorische Algorithmen in C, Skript, Uni Bayreuth, 1997, pp. 5-7 </a> [Broken link]
%H Wolfdieter Lang, <a href="/A139365/a139365.txt">First 6 rows. Factorial representations or Lehmer code for permutations</a>.
%H D. N. Lehmer, <a href="http://dx.doi.org/10.1090/S0002-9904-1906-01419-7">On the orderly listing of substitutions</a>, Bull. AMS 12 (1906), 81-84.
%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>
%F Row n >= 1: sum(facrep(n,m)[n-j],j=1..n), m=0,1,...,n!-1, with the factorial representation facrep(n,m) of m for given n.
%e n=3: The Lehmer codes for the permutations of {1,2,3} are [0,0,0], [0,1,0], [1,0,0], [1,1,0], [2,0,0] and [2,1,0]. These are the factorial representations for 0,1,...,5=3!-1. Therefore row n=3 has the digit sums 0,1,1,2,2,3, the number of inversions of the permutations [1,2,3], [1,3,2], [2,1,3], [2,3,1], [3,1,2] and [3,2,1] (lexicographic order).
%t nn = 5; m = 1; While[Factorial@ m < nn! - 1, m++]; m; Table[Total@ IntegerDigits[k, MixedRadix[Reverse@ Range[2, m]]], {n, 0, 5}, {k, 0, n! - 1}] // Flatten (* Version 10.2, or *)
%t f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], Times @@ Range[# - i]]], {i, 0, #}] &@ NestWhile[# + 1 &, 0, Times @@ Range[# + 1] <= n &]; Most@ Rest[a][[All, 1]]]; Table[Total@ f@ k, {n, 0, 5}, {k, 0, n! - 1}] // Flatten (* _Michael De Vlieger_, Aug 29 2016 *)
%Y Cf. A034968. - _Franklin T. Adams-Watters_, May 13 2009
%Y Cf. A008302.
%K nonn,base,easy,tabf
%O 0,8
%A _Wolfdieter Lang_, May 21 2008
%E Zero term added by _Franklin T. Adams-Watters_, May 13 2009
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