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A062870
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Number of permutations of degree n with greatest sum of distances.
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3
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1, 1, 1, 3, 4, 20, 36, 252, 576, 5184, 14400, 158400, 518400, 6739200, 25401600, 381024000, 1625702400, 27636940800, 131681894400, 2501955993600, 13168189440000, 276531978240000, 1593350922240000, 36647071211520000, 229442532802560000, 5736063320064000000
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OFFSET
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0,4
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COMMENTS
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Number of possible values is 1,2,3,5,7,10,13,17,21,... which I conjecture to be A033638. Maximum distance divided by 2 is the same minus one, i.e., 0,1,2,4,6,9,12,16,20,... which seems to be A002620.
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LINKS
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Georg Fischer, Table of n, a(n) for n = 0..506 (first 301 terms from Alois P. Heinz)
Max Alekseyev, Proof of conjecture
T. Kyle Petersen and Bridget Eileen Tenner, The depth of a permutation, arXiv:1202.4765 [math.CO], 2012.
T. Kyle Petersen and Bridget Eileen Tenner, The depth of a permutation, Journal of Combinatorics 6 (2015), pp. 145--178.
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FORMULA
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a(n) = (n/2)!^2 if n is even else n*((n-1)/2)!^2, cf. A092186. - Conjectured by Vladeta Jovovic, Aug 21 2007; proved (see the link) by Max Alekseyev, Aug 21 2007
a(n) = A062869(n,floor(n^2/4)) for n>=1. - Alois P. Heinz, Oct 02 2022
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EXAMPLE
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(4,3,1,2) has distances (3,1,2,2), sum is 8 and there are 3 other permutations of degree 4 {3, 4, 1, 2}, {3, 4, 2, 1}, {4, 3, 2, 1} with this sum which is the maximum possible.
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MAPLE
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a:= proc(n) option remember; `if`(n<2, 1+n*(n-1),
(n*((n-1)^2*(3*n-4)*a(n-2)-4*a(n-1)))/(4*(n-1)*(3*n-7)))
end:
seq(a(n), n=0..30); # Alois P. Heinz, Jan 16 2014
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MATHEMATICA
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a[n_?EvenQ] := (n/2)!^2; a[n_?OddQ] := n*((n-1)/2)!^2; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 15 2015 *)
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PROG
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(PARI) for(k=0, 20, print1((2*k+1)*k!^2", "(k+1)!^2", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 27 2007
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CROSSREFS
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Cf. A002620, A062866, A062867, A062869.
A007590(n) is the greatest sum of distances for a permutation of degree n. - Dmitry Kamenetsky, Nov 14 2017
Sequence in context: A047165 A124631 A262033 * A226964 A222763 A306668
Adjacent sequences: A062867 A062868 A062869 * A062871 A062872 A062873
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KEYWORD
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nonn
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AUTHOR
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Olivier Gérard, Jun 26 2001
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EXTENSIONS
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a(10)-a(14) from Hugo Pfoertner, Sep 23 2004
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 27 2007
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STATUS
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approved
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