Timeline for Number of integer combinations $x_1 < \cdots < x_n$?
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
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| Jan 19, 2012 at 0:32 | comment | added | Ira Gessel | The formula that Pemantle and Wilf attribute to Proctor is equivalent to a much older formula: the number of paths in the plane from the origin to the point $(a,b)$, where $a > pb$, that stay strictly below the line $x=py$, with steps $(1,0)$ and $(0,1$), is $$\frac{a-pb}{a+b}\binom{a+b}{a}.$$ This formula was apparently first stated by E. Barbier in 1887. A reference is Marc Renault, Four Proofs of the Ballot Theorem, Mathematics Magazine 80 (2007), 345--352; available online at webspace.ship.edu/msrenault/ballotproblem/…. | |
| Jan 18, 2012 at 21:48 | vote | accept | Ralph | ||
| Jan 18, 2012 at 21:39 | comment | added | Ralph | Thanks for the link. Proctor's formula in the linear case is really helpful. | |
| Jan 18, 2012 at 14:31 | history | answered | William J. Keith | CC BY-SA 3.0 |