A368028 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.

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

Offset

0,8

Links

Table of n, a(n) for n=0..77.

math.stackexchange, Passenger entrance/exit combinations

Formula

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).

Example

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.

Crossrefs

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

Sequence in context: A156603 A156612 A351791 * A342645 A350024 A096801

Adjacent sequences: A368025 A368026 A368027 * A368029 A368030 A368031

Keyword

nonn,tabl

Author

Henry Bottomley, Dec 24 2023

Status

approved