RiordanPrize

THE JOHN RIORDAN PRIZE

In 2015 the OEIS Foundation (https://oeisf.org) is offering a prize of $1000 for the best solution to an open problem in the OEIS.

The On-Line Encyclopedia of Integer Sequences invites you to solve an open problem in an entry in the OEIS (https://oeis.org). After adding your solution to the OEIS entry, notify secretary@oeisf.org with subject line “Riordan Prize Nomination”. (It is perfectly acceptable to nominate your own work.) The deadline for submission is December 1, 2015. The decision will be made by a special prize committee, and will be announced at the Joint Mathematics Meetings in Seattle in January 2016.

To find problems to work on, search in the OEIS for the words “conjecture”, “empirical”, “evidence suggests”, “it would be nice”, “would like”, etc. Or find a formula or recurrence (with proof, of course) for a sequence that currently has no formula.

The prize is named after John Riordan (1903–1988; Bell Labs, 1926–1968), author of the classic books An Introduction to Combinatorial Analysis (1958) and Combinatorial Identities (1968), which were the source for hundreds of early entries in the OEIS.

Here is an example from 2008: A145855 gives the number of n-element subsets of {1, ..., 2n-1} whose sum is a multiple of n (1, 1, 4, 9, 26, 76, 246, . . . for n ≥ 1). Vladeta Jovovic conjectured that the nth term is (1/n) * Sum_{d divides n} (−1)^(n+d) phi(n/d) binomial(2d,d), and Max Alekseyev found a proof. (This is not a candidate for the prize, however, since only work carried out in 2015 is eligible.)

The following is a recent unsolved problem, a conjecture of Alois P. Heinz (see A216368). Consider the number of values taken by the nth derivative of x ↑ x ↑ . . . ↑ x (with n x’s and parentheses inserted in all possible ways, and where the up-arrow indicates exponentiation), evaluated at x = 1. Show that this is given by 1, 1, 2, 4, 9, 20, . . ., the number of rooted trees on n nodes (A000081).

Your proof should be mentioned in the appropriate OEIS entry, but (especially if it is long) may be published elsewhere, for example, on the arXiv.

As with anything included in the OEIS, your proof needs to meet the rigorous standards of academic mathematics. Introductions to the format of mathematical proof are provided by Michael Hutchings of the University of California, Berkeley and James Hurley of the University of Connecticut.

In the unlikely event that no suitable submission is received, the committee may decide not to present the award in 2015.

We hope that this competition will lead to the resolution of many open questions in the OEIS!

This page was created on January 28, 2015, by Neil J. A. Sloane, President of the OEIS Foundation.

January 8, 2016: Owing to events beyond our control, there will be a slight delay in announcing the winner of the 2015 Prize. The winner will be announced here within the next two weeks. - N. J. A. Sloane 08:53, 8 January 2016 (UTC)

January 20 2016. Max A. Alekseyev of George Washington University will receive the 2015 John Riordan Prize from the OEIS Foundation for the results in his paper On Enumeration of Dyck--Schröder Paths (arXiv:1601.06158, 2016; JCMCC 106:59-68, 2018). The John Riordan Prize comes with a $1000 prize from the OEIS Foundation.

The classical Catalan and Large Schröder numbers respectively count below-diagonal paths from (0,0) to (n,n) in two classes of directed graphs. In his paper, Max Alekseyev counts paths in a graph that looks like the Catalan graph below the diagonal and the Schröder graph above the diagonal. From this he is able to find generating functions for 22 sequences (A026769-A026779, A026780-A026790), which had been in the On-Line Encyclopedia of Integer Sequences (the OEIS, https://oeis.org) for sixteen years without any formula being discovered.

The 2015 John Riordan Prize was offered for the best solution in 2015 to an open problem in the OEIS. Twenty-three nominations were received, and the decision was not an easy one. The deciding factor in awarding the prize to Max Alekseyev was the unexpected nature of his result (this was much more than finding a proof for an already-conjectured formula), the number of sequences to which it could be applied, and the number of years they had been in the OEIS.

Because the nominations for the 2015 John Riordan Prize arrived by email, and there was no official form to be filled out, the submissions were made in different styles. And some of them used quite personal language. Furthermore, one person sent in multiple nominations for his own work (which was of course permitted). For all these reasons, we decided not to make the list of nominations public.

This was the first time we offered the John Riordan Prize. Next time (if we decide to do it again, which is not clear), we will do things slightly differently. We will provide an official form for submitting a nomination, and we will clearly state in advance that the list of nominations will be made public.