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A126972
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Number of distinct values taken by the entropy for permutations of [1..n], where the entropy of a permutation pi is Sum_{k=1..n} (pi(k)-k)^2.
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4
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1, 1, 2, 4, 11, 21, 36, 57, 85, 121, 166, 221, 287, 365, 456, 561, 681, 817, 970, 1141, 1331, 1541, 1772, 2025, 2301, 2601, 2926, 3277, 3655, 4061, 4496, 4961, 5457, 5985, 6546, 7141, 7771, 8437, 9140, 9881, 10661, 11481, 12342, 13245, 14191, 15181, 16216
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OFFSET
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0,3
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COMMENTS
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Also, number of distinct values taken by sum(k=1..n, k * pi(k) ). - Joerg Arndt, Apr 22 2011
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LINKS
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FORMULA
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G.f.: -(x^7-4*x^6+6*x^5-4*x^4+2*x^3-4*x^2+3*x-1)/(x-1)^4. - M. F. Hasler, Jan 12 2012
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EXAMPLE
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For 24 permutations of {1,2,3,4}, the set of sum(k=1..n, (pi(k)-k)^2) yields {0,2,4,6,8,10,12,14,16,18,20} (11 distinct values).
For 120 permutations of {1,2,3,4,5}, the set of sum(k=1..n, (pi(k)-k)^2) yields {0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,36,38,40} (21 values).
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MATHEMATICA
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LinearRecurrence[{4, -6, 4, -1}, {1, 1, 2, 4, 11, 21, 36, 57}, 50] (* Harvey P. Dale, Jun 01 2016; a(0)=1 prepended by Georg Fischer, Apr 10 2019 *)
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PROG
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(PARI) /* the following inefficient code is for illustrative purpose only: */ A126972(n)={my(u=0, v=vector(n, i, i), t); sum(k=1, n!, !bittest(u, t=norml2(numtoperm(n, k)-v)) & u+=1<<t) } /* M. F. Hasler, Jan 29 2012 */
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Jeff Boscole (jazzerciser(AT)hotmail.com), Mar 20 2007
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EXTENSIONS
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Formula corrected by Joel Brewster Lewis, Aug 18 2009.
Terms corrected, more terms added, and definition clarified by Joerg Arndt, Apr 22 2011.
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STATUS
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approved
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