A100711 Table read by antidiagonals: T(m,n) gives the ordinal number in the table of permutations of length n+1 of the permutation which reverses the first m+1 items on a list of length n+1, leaving the remaining items unaltered. For example, T(5,7) is 28494 and the 28494th row of the permutation table of order 8 is 5 4 3 2 1 0 6 7.

1, 2, 0, 6, 5, 0, 24, 14, 0, 0, 120, 54, 23, 0, 0, 720, 264, 86, 0, 0, 0, 5040, 1560, 414, 119, 0, 0, 0, 40320, 10800, 2424, 566, 0, 0, 0, 0, 85680, 16680, 3294, 719, 0, 0, 0, 131760, 22584, 4166, 0, 0, 0, 177960, 28494, 5039, 0, 0, 224184, 34406, 0, 0, 270414, 40319, 0

Offset

1,2

Comments

The first 8 rows and columns of T are:

1 2 6 24 120 720 5040 40320

0 5 14 54 264 1560 10800 85680

0 0 23 86 414 2424 16680 131760

0 0 0 119 566 3294 22584 177960

0 0 0 0 719 4166 28494 224184

0 0 0 0 0 5039 34406 270414

0 0 0 0 0 0 40319 316646

0 0 0 0 0 0 0 362879

Links

Table of n, a(n) for n=1..61.

Formula

T(m, n) is given by the dot product of (m, m-1, m-2, ..., 1) and (n!, (n-1)!, ..., (1+n-m)!).

Example

T(5,7) is 28494 because we can write (5,4,3,2,1) dot (7!,6!,5!,4!.3!), or (5,4,3,2,1) dot (5040,720,120,24,6) or 28494.

Prog

In Iverson's J language, the program can be written functionally, in Backus's sense: F100711 =: ( [: (- i.) [ ) ( +/ .* ! ) ] ( - i. ) [ ) (- i. 5) is 5 4 3 2 1; (7 (- i.) 5) is 7 6 5 4 3 5 4 3 2 1 (+ / . * !) 7 6 5 4 3 is the dot product of the left argument with the factorials of the right argument, or 28494. So 5 F100711 7 will be 28494.

Crossrefs

See A100630 for another version.

Sequence in context: A078037 A088508 A142457 * A199464 A189961 A211241

Adjacent sequences: A100708 A100709 A100710 * A100712 A100713 A100714

Keyword

easy,nonn,tabl

Author

Eugene McDonnell (eemcd(AT)mac.com), Dec 10 2004

Status

approved