All Questions
Tagged with bijective-combinatorics co.combinatorics
10 questions with no upvoted or accepted answers
14
votes
0
answers
270
views
A symmetry of lattice paths
The number of $n$-step NSEW lattice paths from $(0,0)$ to $(a,b)$ that intersect the line $y=k$ precisely $t$ times is independent of $k$, for $0\leq k\leq b$, where we assume $b\geq0$ for simplicity.
...
- 2,896
10
votes
0
answers
349
views
A bijective proof for the odd companion to Shapiro's Catalan convolution
Shapiro's Catalan convolution is the following formula (where $C_n$ is the $n$th Catalan number):
$$
\sum_{k=0}^{n}{C_{2k}C_{2(n-k)}}=4^nC_n.
$$
In other words, letting $C(z)=\sum_{n=0}^{\infty}{C_nz^...
- 1,190
9
votes
0
answers
180
views
Are there fast rank and unrank algorithms for integer vectors under the action of a permutation group?
We are distributing $m$ indistinguishable balls in $k$ numbered boxes $S=\{1,2,\ldots,k\}$. A distribution is a tuple of nonnegative integers $a=(a_1,\ldots,a_k)$ whose sum is $m$. We also have a ...
- 4,219
8
votes
0
answers
260
views
Efficient listing of ASMs
Famously, the alternating sign matrix theorem gives a product formula for the number $a(n)$ of ASMs of size $n$. There are multiple proofs of this formula, all somewhat involved. My question is ...
- 17k
6
votes
0
answers
188
views
Natural bijection between join- and meet-irreducibles in modular lattices?
A well known property of finite modular lattices is that they have the same number of join-irreducible and meet-irreducible elements. I was wondering if there exists a natural bijection between these ...
- 3,513
6
votes
0
answers
381
views
Reference request: colored Motzkin path interpretation of Catalan numbers
Recall that a Dyck path of length $2n$ is a lattice path in $\mathbb{Z}^2$ from $(0,0)$ to $(2n,0)$ consisting of $n$ up steps $U=(1,1)$ and $n$ down steps $D=(1,-1)$ which never goes below the $x$-...
- 24.2k
6
votes
0
answers
115
views
Distribution of peaks in permutations, after a sorting operation
Let $S_n$ be the set of permutations on $\{1,2,\dotsc,n\}$.
A peak-value of $\pi$ is some $\pi_i$ such that $\pi_{i-1} < \pi_i > \pi_{i+1}$, where $1<i<n$. Let $PV(\pi)$ denote the set of ...
- 15.8k
4
votes
0
answers
225
views
For a combinatorial proof of a symmetric identity
In my paper Supercongruences involving dual sequences [Finite Fields Appl. 46(2017), 179-216], I gave a new symmetric identity which states that if $x+y=-1$ then
$$\sum_{k=0}^n(-1)^k\binom xk^2\binom{...
- 15.6k
2
votes
0
answers
413
views
A (really!) cute identity between product of binomials
As an off-shot of my earlier MO question, I have found a "really cute" identity. The connection is revealed in the limit $q\rightarrow 1$.
So, I would like to ask:
QUESTION. Is there a ...
- 43.2k
2
votes
0
answers
495
views
Bijective proof of a combinatorial identity: $\sum\limits_{k=0}^n\binom nk^2 \binom k{n-m}=\binom nm \binom{n+m}m$
Identity
\begin{equation}
\sum_{k=0}^n\binom nk^2 \binom k{n-m}=\binom nm \binom{n+m}m \tag{1}
\end{equation}
was used in an answer here. As shown in that answer, (1) easily reduces to
\begin{...
- 128k