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A005284 Number of permutations of (1,...,n) having n-6 inversions (n>=6).

%I A005284 M4178 #30 Jun 29 2017 19:28:39
%S A005284 1,6,27,111,440,1717,6655,25728,99412,384320,1487262,5762643,22357907,
%T A005284 86859412,337879565,1315952428,5131231668,20029728894,78265410550,
%U A005284 306109412100,1198306570554,4694809541046,18407850118383
%N A005284 Number of permutations of (1,...,n) having n-6 inversions (n>=6).
%C A005284 Sequence is a diagonal of the triangle A008302 (number of permutations of (1,...,n) with k inversions; see Table 1 of the Margolius reference). - _Emeric Deutsch_, Aug 02 2014
%D A005284 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.14., p.356.
%D A005284 R. K. Guy, personal communication.
%D A005284 E. Netto, Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, 1927, p. 96.
%D A005284 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005284 G. C. Greubel, <a href="/A005284/b005284.txt">Table of n, a(n) for n = 6..1000</a>
%H A005284 R. K. Guy, <a href="/A000707/a000707_2.pdf">Letter to N. J. A. Sloane with attachment, Mar 1988</a>
%H A005284 B. H. Margolius, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/MARGOLIUS/inversions.html">Permutations with inversions</a>, J. Integ. Seqs. Vol. 4 (2001), #01.2.4.
%H A005284 R. H. Moritz and R. C. Williams, <a href="http://www.jstor.org/stable/2690326">A coin-tossing problem and some related combinatorics</a>, Math. Mag., 61 (1988), 24-29.
%F A005284 a(n) = 2^(2*n-7)/sqrt(Pi*n)*Q*(1+O(n^{-1})), where Q is a digital search tree constant, Q = 0.2887880951... (see A048651). - corrected by _Vaclav Kotesovec_, Mar 16 2014
%e A005284 a(7)=6 because we have 2134567, 1324567, 1243567, 1235467, 1234657 and 1234576.
%p A005284 g := proc(n,k) option remember; if k=0 then return(1) else if (n=1 and k=1) then return(0) else if (k<0 or k>binomial(n,2)) then return(0) else g(n-1,k)+g(n,k-1)-g(n-1,k-n) end if end if end if end proc; seq(g(j+6,j),j=0..30); # Barbara Haas Margolius, May 31 2001
%t A005284 Table[SeriesCoefficient[Product[(1-x^j)/(1-x),{j,1,n}],{x,0,n-6}],{n,6,25}] (* _Vaclav Kotesovec_, Mar 16 2014 *)
%Y A005284 Cf. A008302, A000707, A001892, A001893, A001894, A005283, A005285, A048651.
%K A005284 nonn
%O A005284 6,2
%A A005284 _N. J. A. Sloane_
%E A005284 More terms, asymptotic formula from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), May 31 2001
%E A005284 Definition clarified by _Emeric Deutsch_, Aug 02 2014