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A139365
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Array of digit sums of factorial representation of numbers 0,1,...,n!-1 for n>=1.
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1
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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; internal format)
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OFFSET
| 0,8
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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 number of inversions of the permutations arranged in lexicographic order.
Row n is the first n! terms of A034968. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), May 13 2009]
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REFERENCES
| D. N. Lehmer, On the orderly listing of substitutions, Bull. AMS 12 (1906) 81-84.
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LINKS
| A. Kohnert, Kombinatorische Algorithmen in C, Skript, Uni Bayreuth, 1997, pp. 5-7
W. Lang, First 6 rows. Factorial representations or Lehmer code for permutations.
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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.
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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).
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CROSSREFS
| Cf. A034968. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), May 13 2009]
Sequence in context: A073438 A187096 A160115 * A071479 A182631 A091426
Adjacent sequences: A139362 A139363 A139364 * A139366 A139367 A139368
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KEYWORD
| nonn,base,easy,tabf
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) May 21 2008
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EXTENSIONS
| In %H '.' -> 'or','with' -> 'for' In %D changed http link address. - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Sep 09 2008
Zero term added by Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), May 13 2009
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