

A139365


Array of digit sums of factorial representation of numbers 0,1,...,n!1 for n >= 1.


3



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, 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, 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
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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. AdamsWatters, 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. 57 [Broken link]
W. Lang, First 6 rows. Factorial representations or Lehmer code for permutations.
D. N. Lehmer, On the orderly listing of substitutions, Bull. AMS 12 (1906), 8184.
Index entries for sequences related to factorial base representation


FORMULA

Row n >= 1: sum(facrep(n,m)[nj],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

Cf. A034968.  Franklin T. AdamsWatters, May 13 2009
Cf. A008302.
Sequence in context: A340143 A255920 A160115 * A071479 A257398 A182631
Adjacent sequences: A139362 A139363 A139364 * A139366 A139367 A139368


KEYWORD

nonn,base,easy,tabf


AUTHOR

Wolfdieter Lang May 21 2008


EXTENSIONS

In %H '.' > 'or','with' > 'for' In %D changed http link address.  Wolfdieter Lang, Sep 09 2008
Zero term added by Franklin T. AdamsWatters, May 13 2009


STATUS

approved



