The Annals of Mathematical Statistics

The Enumeration of Election Returns by Number of Lead Positions

John Riordan

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In an election return with two candidates $A$ and $B$, if $\alpha_r$ is the number of votes for $A$ in the first $r$ counted, $\beta_r$ the similar number for $B$, then $r$ is a $c$-lead position for $A$ if $\alpha_r > \beta_r + c - 1$. With final vote $(n, m) (n$ for $A, m$ for $B$), what is the number $l_j(n, m; c)$ of returns with $j c$-lead positions? Or, what is the enumerator $l_{n m}(x; c) = \sum l_j(n, m; c)x^j$ of election returns by number of lead positions? For $c = 0, \pm 1, \pm 2, \cdots$ it is shown that all enumerators may be expressed in terms of $l_{n m}(x; 0)$ and $l_{n m}(x; 1)$, which are given explicit expression.

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Ann. Math. Statist., Volume 35, Number 1 (1964), 369-379.

First available in Project Euclid: 27 April 2007

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Riordan, John. The Enumeration of Election Returns by Number of Lead Positions. Ann. Math. Statist. 35 (1964), no. 1, 369--379. doi:10.1214/aoms/1177703760.

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