login
Square array read by antidiagonals; T(n,k) = number of ways a vehicle with capacity k can transport n distinct individuals with distinct starting and finishing points.
0

%I #23 Dec 23 2023 21:50:17

%S 1,0,1,0,1,1,0,2,1,1,0,6,6,1,1,0,24,54,6,1,1,0,120,648,90,6,1,1,0,720,

%T 9720,1944,90,6,1,1,0,5040,174960,52920,2520,90,6,1,1,0,40320,3674160,

%U 1730160,99000,2520,90,6,1,1,0,362880,88179840,65998800,4806000,113400,2520,90,6,1,1,0,3628800,2380855680,2877275520,274050000,6966000,113400,2520,90,6,1,1

%N Square array read by antidiagonals; T(n,k) = number of ways a vehicle with capacity k can transport n distinct individuals with distinct starting and finishing points.

%H math.stackexchange, <a href="https://math.stackexchange.com/a/4832650/6460">Passenger entrance/exit combinations</a>

%F If f(n,k,c)=n*f(n-1,k,c+1)+c*f(n,k,c-1) with f(n,k,c)=0 when n<0 or k<0 or c<0 or k<c and starting with f(0,k,0)=1, then this shows the values of f(n,k,0).

%e T(3,2)=54 represented by the nine patterns AABBCC, AABCBC, AABCCB, ABABCC, ABACBC, ABACCB, ABBACC, ABBCAC, ABBCCA multiplied by 3!=6 for the permutations of A,B,C; but for example ABCABC would not work as the vehicle would be over its capacity of 2 after picking up 3 passengers.

%Y Cf. A080934. Rows include A000012, A057427. Columns include A000007, A000142, A034001. Diagonals include A000680 and A071798.

%K nonn,tabl

%O 0,8

%A _Henry Bottomley_, Dec 24 2023