A100630 Array read by antidiagonals: T(m,n) = Sum_{i=1..m} i*(n-1+i)!.

1, 2, 5, 6, 14, 23, 24, 54, 86, 119, 120, 264, 414, 566, 719, 720, 1560, 2424, 3294, 4166, 5039, 5040, 10800, 16680, 22584, 28494, 34406, 40319, 40320, 85680, 131760, 177960, 224184, 270414, 316646, 362879, 362880, 766080, 1174320, 1583280

Offset

1,2

Comments

Inversion vector corresponding to T(m,n): ( n zeros , 1,2,3,...,m , zeros... )

These are the numbers of permutations (in reverse colexicographical order, compare A055089) that reverse a set of consecutive elements and leave all other elements unchanged. Permutation A(m,n) reverses all elements from n to m+n.

The former title of this sequence refers to finite tables of permutations in lexicographical order: "Triangle read by rows: row n gives the index number in the tables of permutations of order n+1, n+2, ... of the permutation in which the first n items are reversed and the remaining items are in order."

Links

Tilman Piesk, Table of n, a(n) for n = 1..2016

Tilman Piesk, Arrays of permutations (Wikiversity)

Example

T(3,2) = Sum( 1 <= i <= 3 ) [ i * (1+i)! ]
  = 1*(1+1)! + 2*(1+2)! + 3*(1+3)!
  = 1*2 + 2*6 + 3*24
  = 86.
Array begins:
======================================================
m\n |    1     2      3       4        5         6 ...
----+-------------------------------------------------
  1 |    1     2      6      24      120       720 ...
  2 |    5    14     54     264     1560     10800 ...
  3 |   23    86    414    2424    16680    131760 ...
  4 |  119   566   3294   22584   177960   1583280 ...
  5 |  719  4166  28494  224184  1992360  19727280 ...
  6 | 5039 34406 270414 2401464 23765160 259228080 ...
   ...

Prog

Using Iverson's J language, A. p for a permutation p gives the row number in the table of permutations of order (length of p) which has p as its value. For example, q 0 4 9 6 7 5 1 11 8 10 2 3 A. q 13610272.
(PARI) T(m, n) = sum(i=1, m, i*(n-1+i)!) \\ Andrew Howroyd, Oct 05 2025

Crossrefs

See A100711 for another version. Row 2 is A052649.

Sequence in context: A211369 A006596 A000092 * A275839 A057302 A237352

Adjacent sequences: A100627 A100628 A100629 * A100631 A100632 A100633

Keyword

easy,nonn,tabl

Author

Eugene McDonnell (eemcd(AT)mac.com), Dec 03 2004

Extensions

Rewritten by Tilman Piesk, Jul 13 2012

Status

approved