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%I #41 Jun 17 2022 13:21:28
%S 1,1,2,4,11,21,36,57,85,121,166,221,287,365,456,561,681,817,970,1141,
%T 1331,1541,1772,2025,2301,2601,2926,3277,3655,4061,4496,4961,5457,
%U 5985,6546,7141,7771,8437,9140,9881,10661,11481,12342,13245,14191,15181,16216
%N 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.
%C Also, number of distinct values taken by sum(k=1..n, k * pi(k) ). - _Joerg Arndt_, Apr 22 2011
%C For n>=4, sum(k=1..n, k * pi(k) ) takes every value in the interval [A000292(n),A000330(n)] (cf. A175929). - _Max Alekseyev_, Jan 28 2012
%H J. Sack and H. Úlfarsson, <a href="http://arxiv.org/abs/1106.1995">Refined inversion statistics on permutations</a>, arXiv preprint arXiv:1106.1995 [math.CO], 2011-2012.
%H Zhi-Wei Sun, <a href="https://arxiv.org/abs/1811.10503">On permutations of {1, ..., n} and related topics</a>, arXiv:1811.10503 [math.CO], 2018.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F For n>=4, a(n) = 1 + binomial(n+1,3) = 1 + A000330(n) - A000292(n) = 1 + A000292(n-1).
%F 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
%e 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).
%e 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).
%t 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 *)
%o (PARI) A126972(n)=(n!=3)+binomial(n+1,3) \\ - _M. F. Hasler_, Jan 29 2012
%o (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 */
%Y Cf. A007920 (largest permutation entropy), A000292 (average permutation entropy), A135298, A175929.
%K nonn,easy
%O 0,3
%A Jeff Boscole (jazzerciser(AT)hotmail.com), Mar 20 2007
%E Formula corrected by Joel Brewster Lewis, Aug 18 2009.
%E Terms corrected, more terms added, and definition clarified by _Joerg Arndt_, Apr 22 2011.
%E a(0)=1 prepended by _Alois P. Heinz_, Jan 22 2019
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