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 A062870 Number of permutations of degree n with greatest sum of distances. 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..300 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. FORMULA 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 EXAMPLE (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. MAPLE 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 MATHEMATICA 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 *) PROG (PARI) for(k=0, 20, print1((2*k+1)*k!^2", "(k+1)!^2", ")) - Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 27 2007 CROSSREFS Cf. A062866, A062867, A062869, A002620. Sequence in context: A047165 A124631 A262033 * A226964 A222763 A151419 Adjacent sequences:  A062867 A062868 A062869 * A062871 A062872 A062873 KEYWORD nonn AUTHOR Olivier Gérard, Jun 26 2001 EXTENSIONS a(10)-a(14) from Hugo Pfoertner, Sep 23 2004 More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 27 2007 STATUS approved

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