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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 (list; graph; refs; listen; history; text; internal format)
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]

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), 81-84.

Index entries for sequences related to factorial base representation

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

Cf. A034968. - Franklin T. Adams-Watters, 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. Adams-Watters, May 13 2009

STATUS

approved

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Last modified December 5 02:30 EST 2022. Contains 358572 sequences. (Running on oeis4.)