# k-noncrossing and k-nonnesting graphs and fillings of ferrers diagrams

@article{Mier2007knoncrossingAK, title={k-noncrossing and k-nonnesting graphs and fillings of ferrers diagrams}, author={Anna de Mier}, journal={Combinatorica}, year={2007}, volume={27}, pages={699-720} }

We give a correspondence between graphs with a given degree sequence and fillings of Ferrers diagrams by nonnegative integers with prescribed row and column sums. In this setting, k-crossings and k-nestings of the graph become occurrences of the identity and the antiidentity matrices in the filling. We use this to show the equality of the numbers of k-noncrossing and k-nonnesting graphs with a given degree sequence. This generalizes the analogous result for matchings and partition graphs of…

## 49 Citations

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- 2013

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## References

SHOWING 1-10 OF 15 REFERENCES

### Growth diagrams, and increasing and decreasing chains in fillings of Ferrers shapes

- MathematicsAdv. Appl. Math.
- 2006

### Crossings and nestings of matchings and partitions

- Mathematics
- 2005

We present results on the enumeration of crossings and nestings for matchings and set partitions. Using a bijection between partitions and vacillating tableaux, we show that if we fix the sets of…

### Distribution of Crossings, Nestings and Alignments of Two Edges in Matchings and Partitions

- MathematicsElectron. J. Comb.
- 2006

The symmetric distribution of the numbers of crossings and nestings in partitions is derived, which generalizes a recent result of Klazar and Noy in perfect matchings by factorizing the involution through bijections between set partitions and some path diagrams.

### On the Symmetry of the Distribution of k-Crossings and k-Nestings in Graphs

- MathematicsElectron. J. Comb.
- 2006

This paper exhibits a class of graphs for which there are as many $k-noncrossing $2$-nonnesting graphs as $k$- nonnesting $2- noncrossing graphs.

### Generalized triangulations and diagonal-free subsets of stack polyominoes

- MathematicsJ. Comb. Theory, Ser. A
- 2005

### Decreasing Subsequences in Permutations and Wilf Equivalence for Involutions

- Mathematics
- 2004

In a recent paper, Backelin, West and Xin describe a map φ* that recursively replaces all occurrences of the pattern k... 21 in a permutation σ by occurrences of the pattern (k−1)... 21 k. The…

### A spherical initial ideal for Pfaffians

- Mathematics
- 2006

We determine a term order on the monomials in the variables X-ij, 1 <= i <= j <= n, such that corresponding initial ideal of the ideal of Pfaffians of degree r of a generic n by n skew-symmetric…

### A New Class of Wilf-Equivalent Permutations

- Mathematics
- 2001

For about 10 years, the classification up to Wilf equivalence of permutation patterns was thought completed up to length 6. In this paper, we establish a new class of Wilf-equivalent permutation…

### Counting Pattern-free Set Partitions II: Noncrossing and Other Hypergraphs

- MathematicsElectron. J. Comb.
- 2000

Six conjectures stating that if a (multi)hypergraph H has $n$ vertices and does not contain $p$ then the size of ${\cal H}$ is $O(n)$ and the number of such ${\ cal H}s is £O(c^n)$.