W. Lang, Nov 30 2007

a(n,m) tabl head (triangle) for A134830(n,k) (rank k of n-permutation)


    n\k        1        2        3        4        5        6        7        8        9        10 ...


    0          1        0        0        0        0        0        0        0        0         0
   
    1          1        0        0        0        0        0        0        0        0         0

    2          1        0        1        0        0        0        0        0        0         0

    3          2        1        1        2        0        0        0        0        0         0

    4          6        4        3        2        9        0        0        0        0         0

    5         24       18       14       11        9       44        0        0        0         0

    6        120       96       78       64       53       44      265        0        0         0

    7        720      600      504      426      362      309      265     1854        0         0

    8       5040     4320     3720     3216     2790     2428     2119     1854    14833         0

    9      40320    35280    30960    27240    24024    21234    18806    16687    14833    133496
    .
    .
    .


   First column k=1: (n-1)!
   Main diagonal k=n+1: rencontre numbers (subfactorials) A000166(n).
   Row sums: n!. 
   Alternating row sums: A134831: [1, 1, 2, 0, 12, -26, 312, -1338, 16684, -104994,...].   

   Definition of rank k of a permutation sigma from S_n (symmmetric group of n elements):
   Position k of first fixed point of the permutation if k from 1,...,n and if k=n+1 
   (no fixed point in sigma) then R(n)= A000166(n) (rencontre number).
   E.g. n=4, sigma=(2,1,3,4): k=3. sigma=(2,1,4,3): k=5.


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