A248092 - OEIS

%I #14 Jan 09 2017 02:55:24

%S 0,1,0,2,3,0,5,6,5,0,8,9,10,7,0,13,14,15,14,9,0,18,19,20,21,18,11,0,

%T 25,26,27,28,27,22,13,0,32,33,34,35,36,33,26,15,0,41,42,43,44,45,44,

%U 39,30,17,0,50,51,52,53,54,55,52,45,34,19,0

%N Triangle read by rows: T(n,k) is the largest inversion number of the n-permutations having k cycles.

%D M. Bona, Combinatorics of Permutations. 2nd ed., Chapman and Hall/CRC Press, 2012, Boca Raton, FL. pp. 136, 141.

%H P. H. Edelman, <a href="http://dx.doi.org/10.1016/S0195-6698(87)80031-8">On inversions and cycles in permutations</a>, European J. Combinatorics, 8, 1987, 269-279.

%F T(n,k) = binomial(n,k) - ceiling(n/2) + k if k <= ceiling(n/2); T(n,k) = binomial(n,2) - binomial(2k - n, 2) if k > ceiling(n/2).

%e T(4,3)=5; indeed, the 4-permutations with 3 cycles are 1243, 1324, 1432, 2134, 4231, and 3214, having 1, 1, 3, 1, 5, and 3 inversions, respectively.

%e Triangle starts:

%e   0;

%e   1, 0;

%e   2, 3,  0;

%e   5, 6,  5, 0;

%e   8, 9, 10, 7, 0;

%p T := proc (n, k) if n < k then 0 elif k <= ceil((1/2)*n) then binomial(n, 2)-ceil((1/2)*n)+k else binomial(n, 2)-binomial(2*k-n, 2) end if end proc: for n to 13 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form

%K nonn,tabl

%O 1,4

%A _Emeric Deutsch_, Oct 29 2014