There are 2 pending maps:
Identifier
Name
bounce path
Description
Sends a Dyck path $D$ of length $2n$ to its bounce path.
This path is formed by starting at the endpoint $(n,n)$ of $D$ and traveling west until encountering the first vertical step of $D$, then south until hitting the diagonal, then west again until hitting another vertical step of $D$, etc. until the point $(0,0)$ is reached.
This map is the first part of the zeta map Mp00030zeta map.
This path is formed by starting at the endpoint $(n,n)$ of $D$ and traveling west until encountering the first vertical step of $D$, then south until hitting the diagonal, then west again until hitting another vertical step of $D$, etc. until the point $(0,0)$ is reached.
This map is the first part of the zeta map Mp00030zeta map.
Diff Description
Sends a Dyck path D of length 2n to its bounce path.
This path is formed by starting at the endpoint (n,n) of D and travelling west until encountering the first vertical step of D, then south until hitting the diagonal, then west again to hituntil hitting another vertical step of D, etc. until the point (0,0) is reached.
This map is the first part of the zeta map [[Mp00030]].
This path is formed by starting at the endpoint (n,n) of D and travel
This map is the first part of the zeta map [[Mp00030]].
References
[1] Vatter, V. A family of words counted by the Catalan numbers MathOverflow:131585
[2] Haglund, J. The $q$,$t$-Catalan numbers and the space of diagonal harmonics MathSciNet:2371044
[2] Haglund, J. The $q$,$t$-Catalan numbers and the space of diagonal harmonics MathSciNet:2371044
Sage code
def mapping(D):
return D.bounce_path()
Properties
idempotent, graded
Created
Jan 29, 2020 at 14:21 by FindStatCrew
Updated
Oct 01, 2025 at 19:42 by Milena Loper
Identifier
Mp00000:
Integer partitions
—Decrement the largest parts of a partition⟶
Integer partitions
Description
For partitions with the largest part > 1, we decrement all the largest parts by 1 and increment the other (largest possible) parts to obtain the resulting partition of the same number. The resulting partition may grow in length as needed.
Sage code
def increment_first_part(p,length=None):
'''
Increment the part with the smallest index `i` > 1 in partition `p` when such `i` exists.
When `length` is given it bound the length of the resulting partition. Otherwise, it may become longer than `p`.
'''
q = list(p) + [0]*(0 if length is None else length-len(p))
try:
# find smallest index i>0 such that q[i] can be incremented
i = next(i for i in range(1,len(q)) if q[i]<q[i-1])
q[i] += 1
except StopIteration:
if length is None:
# grow length of the partition
q.append(1)
return Partition(q)
def mapping(p):
if len(p)==0 or p[0]==1: return p
m0 = max(i+1 for i in range(len(p)) if p[i]==p[0]) # multiplicity of part p[0]
q = list(p)
for i in range(m0): q[i] -= 1
for i in range(m0): q = increment_first_part(q)
return Partition(q)
Created
Jul 02, 2025 at 19:45 by Max Alekseyev
Updated
Jul 02, 2025 at 19:45 by Max Alekseyev
2 Comments (hide)
Martin Rubey
4 Jul 10:30
4 Jul 10:30
Hi Max!
Do you have any reference in which context this map appears (possibly implicitly)?
Do you know of any desirable properties the map has?
I am asking, because each map comes with (a rather high) cost: each map is combined with all the other maps in all possible ways, up to a certain weight. Therefore, I want to make sure that the "best" (or "useful", "naturally occurring", etc) version of the map is added.
Max Alekseyev
7 Jul 2:39
7 Jul 2:39
This map participates is the recurrence described in my MO answer: https://mathoverflow.net/q/497101
I'm going to add the corresponding statistics and link it to this map.
1 Oct 19:53