RFE Apr 2026: A318233 F_n
Sean A. Irvine
https://oeis.org/wiki/User:Sean_A._Irvine/Requests_for_Enhancements#Requests_for_Enhancements
Sean.
Gareth McCaughan
Hi,
A318233 has the dubious honor of having the shortest sequence name in the OEIS: F_n.
Could this please be improved?
If I have understood the paper correctly -- it looks to me as if it omits one crucial definition, but it seems clear what that should be -- this sequence can be defined as follows. Say that a permutation p is a _derangement_ if it has no fixed points, and _almost decreasing_ if there are no three indices i<j<k for which p(i)<p(j)<p(k). The _excedance_ of a permutation is the number of i for which p(i)>i. Then F_n is the difference between the number of almost decreasing derangements of {1,...,2n} with odd excedance and the number with even excedance.
The paper conjectures that in fact there are more odd-excedance almost increasing derangements of 2n when n is odd, and more even-excedance ones when n is even, and also that there are the same number of both for derangements of 2n+1. However, one can define F_n without that conjecture. It might be better to define F_n = (number of almost decreasing derangements of 2n whose excedance parity matches n's) - (number whose excedance parity doesn't match n's). Or, if you prefer, (-1)^n sum (-1)^excedances(p), the sum being over almost decreasing derangements of 2n.
(I suspect that this conjecture is rather easy to prove. The actual conjecture in the paper says more.)
The connection with A318232 is that that one is (number of ADDs of n) = sum {p an ADD of n} of 1, and this one is (-1)^n sum {p an ADD of 2n} (-1)^excedances(p). These things appear in the paper in the context of a polynomial G_n(t) = sum {p an ADD of n} t^excedances(p), where A318232(n) is G_n(1) and A318233(n) is (-1)^n G_2n(-1).
Unfortunately for the prospects of a brief but clear sequence name, my terminology above is a bit nonstandard: "almost decreasing" is something I just made up (my argument for the name, of course, being that the stronger condition of never having _two_ indices i<j with p(i)<p(j) characterizes decreasing permutations), and what I have called the "excedance" is usually called the "number of excedances", _an_ excedance being an instance where p(i)>i.
For consistency with the title of A318232, I suggest: "Difference between numbers of odd-excedance and even-excedance 123-avoiding derangements of {1,2,...,2n}". The terms "odd-excedance" and "even-excedance" neither appear in the paper nor are standard terminology, but their meaning should be clear to anyone reading the paper. (But if the title is not longer "F_n" then rather than just "See p. 27" the text should say something like "a(n) is the quantity defined as F_n on page 27.")
My extraordinarily crappy Python code for computing these things agrees with the paper's calculations of the first few F_n, and says that the next one is 60438.
--g