Counting pairs of noncrossing binary paths: A bijective approach

Manes, Kostas; Tasoulas, I.; Sapounakis, A.; Tsikouras, P.

doi:10.1016/j.disc.2018.10.016

Cited by 4 publications

(5 citation statements)

References 12 publications

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“…Indeed, since t m+1,m+1 = 2m + 1, formula ( 6) is a direct consequence of the increasing property of T . Moreover, we note that if (7) holds for either one of the cells (i + 1, j), (i, j + 1) then, using the increasing property of T , it must also hold for the cell (i, j). Thus, it is enough to show (7) for every cell (i, j) which is the rightmost cell in its row as well as the lowest cell in its column, i.e., j = i…”

Section: Bijective Proofsmentioning

confidence: 98%

“…It is easy to check that V is a shifted tableau, with v ij ≥ 2 iff j ≥ m + 1, so that relation (9) holds. Since max(T ) = 2m + 1, we obtain that relation (10) holds too, and by relation (7), we obtain automatically that max(V ) ≤ k. For the reverse, set…”

Section: Bijective Proofsmentioning

confidence: 99%

“…Sapounakis et al [11] studied its sublattice of Dyck paths. Recently, a bijection between comparable pairs of paths of this lattice and Dyck prefixes of odd length has been presented [7].…”

Section: Introductionmentioning

confidence: 99%

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Chains of binary paths and shifted tableaux

Manes¹,

Tasoulas²,

Sapounakis³

et al. 2019

Preprint

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In this paper, a natural bijection between multichains of binary paths and shifted tableaux is presented, and it is used for the enumeration of the chains with maximum length from a given path P to the maximum path 1 |P | . By mapping chains to shifted tableaux, the main formulas given in a recent paper by the authors for the enumeration of the P −1 |P | chains having only small intervals and minimum length are proved, using some new bijections on shifted tableaux.

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Section: Bijective Proofsmentioning

confidence: 98%

Section: Bijective Proofsmentioning

confidence: 99%

See 1 more Smart Citation

Chains of binary paths and shifted tableaux

Manes¹,

Tasoulas²,

Sapounakis³

et al. 2019

Preprint

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“…The sublattice D n of Dyck paths has been studied by several authors (e.g., [2,9]). Manes et al [6] have recently presented a bijection between comparable pairs of paths of this lattice and Dyck prefixes of odd length.…”

Section: Introductionmentioning

confidence: 99%

Chains with Small Intervals in the Lattice of Binary Paths

Tasoulas,

Manes,

Sapounakis

et al. 2019

Preprint

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We call an interval [x, y] in a poset small if y is the join of some elements covering x. In this paper, we study the chains of paths from a given arbitrary (binary) path P to the maximum path having only small intervals. More precisely, we obtain and use several formulas for the enumeration of chains having only small intervals and minimal length. For this, we introduce and study the notions of filling and degree of a path, giving in addition some related statistics.

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“…This lattice appears in the literature in various equivalent forms (e.g., sequences of integers [13], binary words [4, p. 92], subsets of [n] [5], permutations of [n] [15, p. 402], partitions of n into distinct parts [14], threshold graphs [7]). Ferrari and Pinzani [3] and Sapounakis et al [9] have studied its sublattice of Dyck paths, Manes et al have presented a bijection between comparable pairs of paths of this lattice and Dyck prefixes of odd length [6] and Tasoulas et al have studied the chains with small intervals in this lattice [16].…”

mentioning

confidence: 99%

Hamiltonian Intervals in the Lattice of Binary Paths

Tasoulas,

Manes,

Sapounakis

2024

Electron. J. Combin.

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Let $\mathcal{P}_n$ be the set of all binary paths (i.e., lattice paths with upsteps $u = (1,1)$ and downsteps $d = (1,-1)$) of length $n$ endowed with the pointwise partial ordering (i.e., $P \le Q$ iff the lattice path $P$ lies weakly below $Q$) and let $G_n$ be its Hasse graph. For each path $P \in \mathcal{P}_n$, we denote by $I(P)$ the interval which contains the elements of $\mathcal{P}_n$ less than or equal to $P$, excluding the first two elements of $\mathcal{P}_n$, and by $G(P)$ the subgraph of $G_n$ induced by $I(P)$. In this paper, it is shown that $G(P)$ is Hamiltonian iff $P$ contains at least two peaks and $I(P)$ has equal number of elements with even and odd rank. The last condition is always true for paths ending with an upstep, whereas, for paths ending with a downstep, a simple characterization is given, based on the structure of the path.

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Counting pairs of noncrossing binary paths: A bijective approach

Cited by 4 publications

References 12 publications

Chains of binary paths and shifted tableaux

Chains of binary paths and shifted tableaux

Chains with Small Intervals in the Lattice of Binary Paths

Hamiltonian Intervals in the Lattice of Binary Paths

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