sum of n-wide products of permutations

[Eugene McDonnell]

In a message of February 9 on "n-wide permutations sums of products" I 
reported min and max values for 3, 4, 5, 6, 7 and 8-wide overlapping 
infixes for permutations of 3 through 9, for non-cyclic and cyclic 
cases.

When I get a lot of related lists, I think of putting them all in the 
same array. Here are the two noncyclic tables. Row n gives results for 
n-wide products; the column heads give the width of the permutations 
being treated.

    Non-cyclic min
\  1 2 3  4   5   6    7     8     9
1  1 3 6 10  15  21   28    36    45
2    2 5 12  22  38   59    88   124
3      6 14  28  68  123   203   333
4        24  54 100  196   504   924
5           120 264  468   832  1680
6               720 1560  2688  4512
7                   5040 10800 18240
8                        40320 85680

Non-cyclic max
\  1 2 3  4   5    6    7     8      9
1  1 3 6 10  15   21   28    36     45
2    2 9 23  46   80  127   189    268
3      6 36 115  273  546   976   1611
4        24 180  690 1911  4353   8706
5           120 1080 4830 15288  39177
6                720 7560 38640 137592
7                    5040 60480 347760
8                         40320 544320


I just piled the values I had found on each other in what seemed a 
reasonable triangular way, and much to my surprise found that the 
second diagonals of these, except for the first value, are in OEIS 
already:

Min: A052649: 1 3 9 36 180 1080 7560 60480 544320, with formula a(n) = 
(3/2)*n!
Max: A070960: 2 5 14 54 264 1560 10800 85680, with formula    a(n) = 
(3+2*n)*n!

I haven't yet found anything like this for the cyclic cases.

If anyone has insight into these, I'd appreciate hearing it.

Eugene McDonnell


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