### Abstract

We study two types of sequences of positive integers which arise from problems in the measurement of comparative judgements of probability. The first type consists of the Van Lier sequences, which are nondecreasing sequences x_{1}, x_{2},...,x_{n} of positive integers that start with two 1's and have the property that, whenever j<k≤n,x_{k}-x_{j} can be expressed as a sum of terms from the sequence other than x_{j}. The second type consists of the regular sequences, which are nondecreasing sequences of positive integers that start with two 1's and have the property that each subsequent term is a partial sum of preceding terms. We show that every regular sequence without "gaps" is Van Lier and that every regular sequence which satisfies the Fibonacci-like inequality x_{k}≤x_{k-2}+x_{k-1} is Van Lier. We also study one-term extensions of Van Lier sequences and obtain some asymptotic results on the number of Van Lier sequences.

Original language | English |
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Pages (from-to) | 209-220 |

Number of pages | 12 |

Journal | Discrete Applied Mathematics |

Volume | 27 |

Issue number | 3 |

DOIs | |

State | Published - 1 Jan 1990 |

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*Discrete Applied Mathematics*,

*27*(3), 209-220. https://doi.org/10.1016/0166-218X(90)90066-L

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*Discrete Applied Mathematics*, vol. 27, no. 3, pp. 209-220. https://doi.org/10.1016/0166-218X(90)90066-L

**Van Lier sequences.** / Fishburn, Peter C.; Roberts, Fred S.; Roberts, Helen M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Van Lier sequences

AU - Fishburn, Peter C.

AU - Roberts, Fred S.

AU - Roberts, Helen M

PY - 1990/1/1

Y1 - 1990/1/1

N2 - We study two types of sequences of positive integers which arise from problems in the measurement of comparative judgements of probability. The first type consists of the Van Lier sequences, which are nondecreasing sequences x1, x2,...,xn of positive integers that start with two 1's and have the property that, whenever jk-xj can be expressed as a sum of terms from the sequence other than xj. The second type consists of the regular sequences, which are nondecreasing sequences of positive integers that start with two 1's and have the property that each subsequent term is a partial sum of preceding terms. We show that every regular sequence without "gaps" is Van Lier and that every regular sequence which satisfies the Fibonacci-like inequality xk≤xk-2+xk-1 is Van Lier. We also study one-term extensions of Van Lier sequences and obtain some asymptotic results on the number of Van Lier sequences.

AB - We study two types of sequences of positive integers which arise from problems in the measurement of comparative judgements of probability. The first type consists of the Van Lier sequences, which are nondecreasing sequences x1, x2,...,xn of positive integers that start with two 1's and have the property that, whenever jk-xj can be expressed as a sum of terms from the sequence other than xj. The second type consists of the regular sequences, which are nondecreasing sequences of positive integers that start with two 1's and have the property that each subsequent term is a partial sum of preceding terms. We show that every regular sequence without "gaps" is Van Lier and that every regular sequence which satisfies the Fibonacci-like inequality xk≤xk-2+xk-1 is Van Lier. We also study one-term extensions of Van Lier sequences and obtain some asymptotic results on the number of Van Lier sequences.

UR - http://www.scopus.com/inward/record.url?scp=45149140816&partnerID=8YFLogxK

U2 - 10.1016/0166-218X(90)90066-L

DO - 10.1016/0166-218X(90)90066-L

M3 - Article

AN - SCOPUS:45149140816

VL - 27

SP - 209

EP - 220

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

IS - 3

ER -