On the number of single-peaked narcissistic or single-crossing narcissistic preference profiles

We investigate preference profiles for a set v of voters, where each voter i has a preference order >(i); on a finite set A of alternatives (that is, a linear order on A) such that for each two alternatives a, b is an element of A, voter i prefers a to 6 if a>(i)b. Such a profile is narcissistic if each alternative a is preferred the most by at least one voter. It is single-peaked if there is a linear order (sic)(SP) on the alternatives such that each voter's preferences on the alternatives along the order (sic)(SP) are either strictly increasing, or strictly decreasing, or first strictly increasing and then strictly decreasing. It is single-crossing if there is a linear order (sic)(SP) on the voters such that each pair of alternatives divides the order (sic)(SP) into at most two suborders, where in each suborder, all voters have the same linear order on this pair. We show that for n voters and n alternatives, the number of single-peaked narcissistic profiles is Pi(n-1)(i=2)((n-1)(i-1)) while the number of single-crossing narcissistic profiles is 2((n-1)(2)). (C) 2018 Elsevier B.V. All rights reserved.