## Abstract

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∈A, voter i prefers a to b 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 ▹^{sp} on the alternatives such that each voter's preferences on the alternatives along the order ▹^{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 ▹^{sc} on the voters such that each pair of alternatives divides the order ▹^{sc} 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 ∏_{i=2}^{n−1}[Formula presented] while the number of single-crossing narcissistic profiles is 2^{[Formula presented]}.

Original language | English |
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Pages (from-to) | 1225-1236 |

Number of pages | 12 |

Journal | Discrete Mathematics |

Volume | 341 |

Issue number | 5 |

DOIs | |

State | Published - 1 May 2018 |

## Keywords

- Narcissistic preferences
- Semi-standard Young tableaux
- Single-crossing preferences
- Single-peaked preferences

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics