n-wide permutation sums of products

[Eugene McDonnell]

Here's a beginning at giving the minima and maxima for the sums of the 
products of n-wide overlapping infixes of permutations, for 2<n<9. The 
columns headed min index and max index give the 0-offset row numbers of 
the first occurrence of the minimum and maximum in lexically ordered 
tables of permutations. The data look neater if you use an equal-width 
font such as Courier. Results are shown for both the non-cyclic and 
cyclic cases.

For example, the non-cyclic minimum for permutations of length 8 is 
found in row 33384, which is 7 5 3 2 1 4 6 8. The five infixes of 
length 4 are
7 5 3 2
5 3 2 1
3 2 1 4
2 1 4 6
1 4 6 8
Their products are
210 30 24 48 192
Which sum to
504


Non-cyclic
3-wide
             min    max
n  min  max index  index
3    6    6      0    0
4   14   36     12    3
5   28  115     84   11
6   68  273    534   41
7  123  546   3870  191
8  203  976  32550 1055
9  333 1611 303294 6959

4-wide
4   24   24      0    0
5   54  180     72    9
6  100  690    552   41
7  196 1911   4008  191
8  504 4353  33384 1055
9  924 8706 304008 6959

5-wide
5   120   120      0    0
6   264  1080    480   33
7   468  4830   4080  185
8   832 15288  33480 1055
9  1680 39177 309840 6959

6-wide
6   720    720      0    0
7  1560   7560   3600  153
8  2688  38640  33840 1025
9  4512 137592 310320 6935

7-wide
7   5040   5040      0    0
8  10800  60480  30240  873
9  18240 347760 312480 6785

8-wide
8   40320  40320      0    0
9   85680 544320 282240 5913


Cyclic
3-wide
             min    max
n  min  max index  index
3   12   12      0    0
4   26   44      8    1
5   57  123     62    3
6  116  281    536   11
7  192  554   2672   41
8  316  984  23918  191
9  493 1619 308552 1055

4-wide
4   48   48      0    0
5   94  220     60    3
6  174  738    444   11
7  407 1959   1974   41
8  882 4401  24030  191
9 1404 8754 273780 1055

5-wide
5  240   240      0    0
6  444  1320    432    9
7  762  5166   3432   41
8 1482 15672  19128  191
9 3631 39561 114024 1055

6-wide
6 1440   1440      0    0
7 2568   9240   3360   33
8 4212  41328  29280  185
9 7440 141048 235680 1055

7-wide
7 10080  10080      0    0
8 17520  73920  28800  153
9 27864 371952 275760 1025

8-wide
8  80640  80640      0   0
9 137520 665280 272160 873

Eugene McDonnell
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