login
T(n,k) = 0!*1!*2!*...*(k-1)! *(n*k)! *k*(k-1)*n*(n-1) / 2*n!*(n+1)!*...*(n+k-1)!
5

%I #4 Jul 12 2012 00:39:52

%S 0,0,0,0,4,0,0,30,30,0,0,168,756,168,0,0,840,16632,16632,840,0,0,3960,

%T 360360,1729728,360360,3960,0,0,18018,7876440,199536480,199536480,

%U 7876440,18018,0,0,80080,174594420,25241364720,140229804000,25241364720

%N T(n,k) = 0!*1!*2!*...*(k-1)! *(n*k)! *k*(k-1)*n*(n-1) / 2*n!*(n+1)!*...*(n+k-1)!

%C (Empricial) T(n,k)=Number of nXk matrices containing a defective permutation of 1..n*k in strictly increasing order rowwise and columnwise, with one permutation value omitted and one repeated (see example)

%C Formula is n*(n-1)*k*(k-1)/2 times n-th k-dimensional Catalan number

%C Table starts

%C .0.......0.............0....................0...........................0

%C .0.......4............30..................168.........................840

%C .0......30...........756................16632......................360360

%C .0.....168.........16632..............1729728...................199536480

%C .0.....840........360360............199536480................140229804000

%C .0....3960.......7876440..........25241364720.............118949931243000

%C .0...18018.....174594420........3445446284280..........117015012361447200

%C .0...80080....3926434512......500598983364480.......129624266420759510400

%C .0..350064...89492111280....76591644454765440....158211402715245473193600

%C .0.1511640.2064420294300.12237255920840932800.209298196564031904834960000

%H R. H. Hardin, <a href="/A181204/b181204.txt">Table of n, a(n) for n=1..1000</a>

%e Some solutions for 4X2

%e ..2..4....1..4....1..3....1..3....2..5....1..3....1..3....2..4....1..2....1..3

%e ..3..5....2..5....3..6....2..4....3..6....2..4....2..4....3..6....2..4....2..4

%e ..5..7....6..7....4..7....4..6....4..7....4..5....4..7....5..7....3..5....4..7

%e ..6..8....7..8....5..8....5..8....5..8....7..8....5..8....6..8....6..8....6..8

%Y Column 2 is twice A002740(n+1)

%Y Cf. A060854 for permutation without defect.

%K nonn,tabl

%O 1,5

%A _R. H. Hardin_ Oct 10 2010