A342140 Number of permutations of degree n with greatest sum of distances and highest Shannon entropy.

1, 1, 3, 2, 17, 4, 86, 4, 488, 12, 3172, 40, 22912, 56, 166814, 256

Offset: 1

Author

Andrea G. Amato, Mar 01 2021

Keywords

• nonn
• hard
• more

Comments

Starting from a list of n ordered numbers, the sequence gives the number of permutations of the list that display both the greatest sum of distances (see A007590 and A062870) and the highest Shannon entropy (see A341838 for a more in-depth explanation on how to calculate it).

A way to interpret this is to see these permutations as the ones with both the highest level of disorder and the greatest distance from a starting configuration.

Examples

			Starting from (1,2,3,4), there are only two permutations that have both the greatest sum of distances (which is 8 for n=4) and the highest Shannon entropy (which is 1.039720... for n=4). These permutations are (3,4,2,1) and (4,3,1,2).
		

Links

• Andrea G. Amato How well can you shuffle a deck of cards?
• Andrea G. Amato, Conjectures and properties

Crossrefs

Cf. A007590 (greatest sum of distances of a given n).

Cf. A062870 (permutations that possess this property).

Cf. A341838 (number of permutations with the highest Shannon entropy).

Extensions

a(13)-a(15) from Hugo Pfoertner, Mar 02 2021

a(16) from Hugo Pfoertner, Mar 07 2021