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 A126972 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. 4
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Also, number of distinct values taken by sum(k=1..n, k * pi(k) ). - Joerg Arndt, Apr 22 2011 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 LINKS Zhi-Wei Sun, On permutations of {1, ..., n} and related topics, arXiv:1811.10503 [math.CO], 2018. Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1). FORMULA For n>=4, a(n) = 1 + binomial(n+1,3) = 1 + A000330(n) - A000292(n) = 1 + A000292(n-1). 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 EXAMPLE 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). MATHEMATICA 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 *) PROG (PARI) A126972(n)=(n!=3)+binomial(n+1, 3)  \\ - M. F. Hasler, Jan 29 2012 (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<

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Last modified September 28 10:48 EDT 2021. Contains 347714 sequences. (Running on oeis4.)