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A189073 Triangle read by rows: T(n,k) is the number of inversions in k-compositions of n for n >= 3, 2 <= k <= n-1.

%I A189073 #29 Mar 19 2014 15:58:11
%S A189073 1,1,3,2,6,6,2,12,18,10,3,18,42,40,15,3,27,78,110,75,21,4,36,132,240,
%T A189073 240,126,28,4,48,204,460,600,462,196,36,5,60,300,800,1290,1302,812,
%U A189073 288,45,5,75,420,1300,2490,3108,2548,1332,405,55,6,90,570,2000,4440,6594,6692,4608,2070,550,66
%N A189073 Triangle read by rows: T(n,k) is the number of inversions in k-compositions of n for n >= 3, 2 <= k <= n-1.
%C A189073 The Heibach et al. reference has a table for n <= 14.
%H A189073 Alois P. Heinz, <a href="/A189073/b189073.txt">Rows n = 3..143, flattened</a>
%H A189073 S. Heubach, A. Knopfmacher, M. E. Mays and A. Munagi, <a href="http://www.calstatela.edu/faculty/sheubac/papers/inversns_in_composns.pdf ">Inversions in Compositions of Integers</a>, to appear in Quaestiones Mathematicae.
%F A189073 G.f.: (1-x)*x^3/((1+x)*(1-x-y*x)^3). - _Geoffrey Critzer_, Mar 19 2014
%e A189073 Triangle begins:
%e A189073 1;
%e A189073 1,  3;
%e A189073 2,  6,   6;
%e A189073 2, 12,  18,   10;
%e A189073 3, 18,  42,   40,   15;
%e A189073 3, 27,  78,  110,   75,   21;
%e A189073 4, 36, 132,  240,  240,  126,   28;
%e A189073 4, 48, 204,  460,  600,  462,  196,   36;
%e A189073 5, 60, 300,  800, 1290, 1302,  812,  288,   45;
%e A189073 5, 75, 420, 1300, 2490, 3108, 2548, 1332,  405,  55;
%e A189073 6, 90, 570, 2000, 4440, 6594, 6692, 4608, 2070, 550, 66;
%e A189073 ...
%e A189073 T(5,3) = 6 because we have: 3+1+1, 1+3+1, 1+1+3, 2+2+1, 2+1+2, 1+2+2 having 2,1,0,2,1,0 inversions respectively. - _Geoffrey Critzer_, Mar 19 2014
%p A189073 T:= proc(n, k) option remember;
%p A189073       if k=2 then floor((n-1)/2)
%p A189073     elif k>=n then 0
%p A189073     else T(n-1, k) +k/(k-2) *T(n-1, k-1)
%p A189073       fi
%p A189073     end:
%p A189073 seq(seq(T(n, k), k=2..n-1), n=3..13);  # _Alois P. Heinz_, Apr 17 2011
%t A189073 T[n_, k_] := T[n, k] = Which[k == 2, Floor[(n-1)/2], k >= n, 0, True, T[n-1, k] + k/(k-2)*T[n-1, k-1]]; Table[Table[T[n, k], {k, 2, n-1}], {n, 3, 13}] // Flatten (* _Jean-François Alcover_, Jan 14 2014, after _Alois P. Heinz_ *)
%Y A189073 Row sums are A189052. The first column is A004526(n-1). Diagonal is A000217(n-2).  Lower diagonal is A002411(n-3). 2nd lower diagonal is A001621(n-4).
%K A189073 nonn,tabl
%O A189073 3,3
%A A189073 _N. J. A. Sloane_, Apr 16 2011