Hi,
Well I think that dinv would be number_of_diagonal_inversions but I
agree with Hugh, the statistics area(D), bounce(D) and dinv(D) are all
fairly standard (add Haglund's book + dozens of references since) and
certainly way more descriptive than a_statistic, b_statistic and
d_statistic; however the advantage of those descriptors is that they
indicate their equi-distribution over the set of Dyck paths (that is,
\sum_D q^{x(D)} is the q-Catalan for x = a,b,d-statistic).
Just a word about bounce(D) though: it is sort of a major index of the
touch points of the bounce path. There are the obvious involutions on the
touch point set (complementation and reverse) and I think I have seen the
size_of_the_set_of_touch_points_of_the_bounce_path(D) come up in a formula
somewhere.
-Mike
On Thursday, 17 May 2012 12:40:43 UTC-4, Hugh Thomas wrote:
Hi!
I think area, bounce, and dinv are standard terms. Spelling out
what dinv means would only confuse matters for me. I don't remember the
precise definition --- it is complicated. So even spelling outtwo or three
words would still be only be able to give an imprecise description. That
being the case, I think the best thing is to use the standard term.
cheers,
Hugh
On Thursday, May 17, 2012 5:45:19 AM UTC-3, Christian Stump wrote:
Hi,
I am currently working on filling the wiki at www.findstat.org with
more data, since Chris Berg and I are going to present it at the FPSAC
this summer.
I would like to make the following changes to Dyck words:
1. changing a-statistic, b-statistic, d-statistic to area, bounce,
dinv (and leaving the old with a warning).
2. changing the to_partition to return the partition when cutting out
the path above the diagonal (currently, the transposed is returned).
Is that okay with everyone?
Christian
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