OFFSET
0,8
COMMENTS
The row lengths sequence is A000142 (factorials).
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.
Row n is the first n! terms of A034968. - Franklin T. Adams-Watters, May 13 2009
LINKS
Alois P. Heinz, Rows n = 0..8, flattened
FindStat - Combinatorial Statistic Finder, The number of inversions of a permutation
A. Kohnert, Kombinatorische Algorithmen in C, Skript, Uni Bayreuth, 1997, pp. 5-7 [Broken link]
Wolfdieter Lang, First 6 rows. Factorial representations or Lehmer code for permutations.
D. N. Lehmer, On the orderly listing of substitutions, Bull. AMS 12 (1906), 81-84.
FORMULA
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.
EXAMPLE
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).
MATHEMATICA
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 *)
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 *)
CROSSREFS
KEYWORD
nonn,base,easy,tabf
AUTHOR
Wolfdieter Lang, May 21 2008
EXTENSIONS
Zero term added by Franklin T. Adams-Watters, May 13 2009
STATUS
approved