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A simple bijection for enhanced, classical, and 2-distant k -noncrossing partitions
Discrete Mathematics xxx (xxxx) xxx
Contents lists available at ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
A simple bijection for enhanced, classical, and 2-distant k-noncrossing partitions✩ ∗
Juan B. Gil a , , Jordan O. Tirrell b a b
Penn State Altoona, 3000 Ivyside Park, Altoona, PA 16601, USA Department of Mathematics and Computer Science, Washington College, Chestertown, MD 21620, USA
article
info
a b s t r a c t In this note, we give a simple extension map from partitions of subsets of [n] to partitions of [n + 1], which sends δ -distant k-crossings to (δ + 1)-distant k-crossings (and similarly for nestings). This map provides a combinatorial proof of the fact that the numbers of enhanced, classical, and 2-distant k-noncrossing partitions are each related to the next via the binomial transform. Our work resolves a recent conjecture of Zhicong Lin and generalizes earlier reduction identities for partitions. © 2019 Elsevier B.V. All rights reserved.
Article history: Received 3 December 2018 Received in revised form 4 August 2019 Accepted 17 October 2019 Available online xxxx Keywords: k-noncrossing partition Enhanced k-noncrossing partition
1. Introduction Given a partition π of a set of integers and k ≥ 1, the arc diagram of π is obtained by drawing an arc between each pair of integers that appears consecutively in the same block of π . For example, the partition π = {148, 26, 3, 57} can be represented as
1
2
3
4
5
6
7
8
and we say that 1246, 2468 and 2567 are crossings, and 4578 is a nesting. Additionally, we say 148 is an enhanced crossing and 134 is an enhanced nesting. Precisely, an enhanced k-crossing is a sequence a1 < a2 < · · · < ak ≤ b1 < b2 < · · · < bk such that there is an arc between each pair ai , bi , and we consider singleton blocks to have trivial arcs. A classical k-crossing additionally requires that ak < b1 , and in general a δ -distant k-crossing requires that the distance b1 − ak is at least δ . Thus, the enhanced crossings correspond to δ = 0 and the classical crossings correspond to δ = 1. Note that an enhanced 1-crossing is simply an arc, and a classical 1-crossing is a nontrivial arc. An enhanced/classical/δ -distant k-nesting is defined similarly with an arc between each pair ai , bk+1−i . Returning to our example above, there is one enhanced ✩ The authors would like to thank the Department of Mathematics and Statistics and the Hutchcroft fund at Mount Holyoke College for their support. ∗ Corresponding author. E-mail addresses: [email protected] (J.B. Gil), [email protected] (J.O. Tirrell). https://doi.org/10.1016/j.disc.2019.111705 0012-365X/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: J.B. Gil and J.O. Tirrell, A simple bijection for enhanced, classical, and 2-distant k-noncrossing partitions, Discrete Mathematics (2019) 111705, https://doi.org/10.1016/j.disc.2019.111705.
2
J.B. Gil and J.O. Tirrell / Discrete Mathematics xxx (xxxx) xxx Table 1 Sequences NCδ,k (n) for small k and δ = 0, 1, 2. n
0
1
2
3
4
5
6
7
8
9
10
11
OEIS[17]
NC0,1 (n) NC0,2 (n) NC0,3 (n) NC0,4 (n) NC0,5 (n)
1 1 1 1 1
0 1 1 1 1
0 2 2 2 2
0 4 5 5 5
0 9 15 15 15
0 21 51 52 52
0 51 191 203 203
0 127 772 876 877
0 323 3320 4120 4140
0 835 15032 20883 21146
0 2188 71084 113034 115945
0 5798 348889 648410 678012
A001006 A108307 A192855 A192865
NC1,1 (n) NC1,2 (n) NC1,3 (n) NC1,4 (n) NC1,5 (n)
1 1 1 1 1
1 1 1 1 1
1 2 2 2 2
1 5 5 5 5
1 14 15 15 15
1 42 52 52 52
1 132 202 203 203
1 429 859 877 877
1 1430 3930 4139 4140
1 4862 19095 21119 21147
1 16796 97566 115495 115974
1 58786 520257 671969 678530
A000108 A108304 A108305 A192126
NC2,1 (n) NC2,2 (n) NC2,3 (n) NC2,4 (n) NC2,5 (n)
1 1 1 1 1
1 1 1 1 1
2 2 2 2 2
4 5 5 5 5
8 15 15 15 15
16 51 52 52 52
32 188 203 203 203
64 731 876 877 877
128 2950 4115 4140 4140
256 12235 20765 21146 21147
512 51822 111301 115938 115975
1024 223191 627821 677765 678569
A000079 A007317 A318191 A318192 A318193
B(n)
1
1
2
5
15
52
203
877
4140
21147
115975
678570
A000110
3-crossing 12468, and no enhanced 3-nesting. Of the classical crossings and nestings, all but one are 2-distant, and none are 3-distant. Much of the combinatorial interest in these structures follows the work of Chen, Deng, Du, Stanley, and Yan [5]. They proved a beautiful bijective symmetry between k-crossings and k-nestings, which D. Drake & J.S. Kim [6] generalized to δ -distant k-crossings and k-nestings (see Eq. (3)). Motivation for δ -distant k-crossings also comes from the study of RNA structures [8]. Let NCδ,k (n) denote the number of δ -distant k-noncrossing partitions of [n] = {1, . . . , n}, that is, partitions of [n] with no δ -distant k-crossing. Examples are given in Table 1. In this paper, we give combinatorial proofs of the following binomial transform identities. Theorem 1.
For integers n ≥ 0 and k ≥ 1,
NC1,k (n + 1) =
n ( ) ∑ n i=0 n
NC2,k (n + 1) =
i
NC0,k (i) and
∑ ( n) i=0
i
NC1,k (i).
(1)
(2)
Eq. (1) is well-known for k = 2, and it was recently proven for k = 3 by Lin [13], who conjectured it held for all k. In the final version of [13], Lin announced two different combinatorial proofs of his conjecture in( joint ) work with D. Kim [14]. 2n 1 Eq. (2) appears to be new for k > 2. For k = 2, it is well-known that NC1,2 (n) = n+ . The fact that NC2,2 (n) is 1 n the binomial transform of NC1,2 (n) was shown by Drake & J.S. Kim [6] using Charlier diagrams and generating functions. Several bijections with other objects can be found in [7,10]. Bousquet-Mélou & Xin [2] showed that the sequences NC0,3 (n) and NC1,3 (n) are P-recursive and gave explicit recurrences. Mishna & Yen [16] used generating trees to find functional equations for NC1,k (n) when k > 3. Burrill, Elizalde, Mishna, and Yen [3] did the same for NC0,k (n). Theorem 1 provides a simple way to connect the enumerations of NC0,k (n), NC1,k (n), and NC2,k (n). In particular, as Lin [13] observed, the binomial transform preserves D-finiteness (see [18, Theorem 6.4.10]). Corollary 2. The D-finiteness of the ordinary generating functions for NC0,k (n), NC1,k (n), and NC2,k (n) are the same.1 Unfortunately, the sequence NC3,k (n + 1) is not the binomial transform of the sequence NC2,k (n), and in general, not much is known about NCδ,k (n) for δ ≥ 3 (the generating function for NC3,2 (n) is given in [11]). Let NCNδ,k,ε,j (n) denote the number of δ -distant k-noncrossing partitions of [n] which also contain no ε -distant j-nesting. By the celebrated symmetry of Chen et al. [5], generalized by Drake & Kim [6], we have NCNδ,k,ε,j (n) = NCNε,j,δ,k (n).
(3)
We will give a combinatorial proof of the following refinement of Theorem 1. 1 For k > 3, it is conjectured [2] that the generating function of NC (n) is not D-finite. 1 ,k Please cite this article as: J.B. Gil and J.O. Tirrell, A simple bijection for enhanced, classical, and 2-distant k-noncrossing partitions, Discrete Mathematics (2019) 111705, https://doi.org/10.1016/j.disc.2019.111705.
J.B. Gil and J.O. Tirrell / Discrete Mathematics xxx (xxxx) xxx
3
Fig. 1. The arc diagram for π = {1479, 25, 6} extends to πˆ = {15, 267T, 3, 48, 9}.
Theorem 3.
For integers n ≥ 0, j, k ≥ 1, and ε, δ ∈ {0, 1},
NCNδ+1,k,ε+1,j (n + 1) =
n ( ) ∑ n i=0
i
NCNδ,k,ε,j (i)
(4)
The generating function for NCN1,k,1,j (n) is not simply D-finite but in fact rational for all k, j ≥ 1. This was shown by Marberg [15] using a bijection with walks on certain multigraphs (and he actually proved this in the more general setting of colored partitions). By Theorem 3, the generating functions for NCN0,k,0,j (n) and NCN2,k,2,j (n) are also rational. 2. Main bijection It is known that the sequence B(n) of Bell numbers, counts the number of partitions of [n], is an eigensequence ) ∑n (which n of the binomial transform. That is, B(n + 1) = i=0 i B(i). The usual combinatorial proof of this fact uses a bijection from partitions of subsets of [n] to partitions of [n + 1], defined by adding a new block containing n + 1 together with every element of [n] not already contained in a block. For example, if n = 9, this bijection takes {1479, 25, 6} to {1479, 25, 6, 38T}, where T stands for 10. We now give a different bijection which behaves well with respect to crossings and distance. Definition 4. For any n ≥ 0, and for any partition π of some subset of [n], we define the partition πˆ of [n + 1] as follows. For each pair of vertices v < w appearing consecutively in the same block of π , place v and w + 1 in the same block of πˆ . For each singleton u of π , place u and u + 1 in the same block of πˆ . At last, place any remaining vertices in [n + 1] as singletons of πˆ . We will refer to π ↦ → πˆ as the extension map. For example, if n = 9 and π = {1479, 25, 6}, then πˆ = {15, 267T, 3, 48, 9}. An elegant geometric description of this map can be obtained by extending the lines in the arc diagram of π to arrive at the arc diagram of πˆ (see Fig. 1), where singletons are considered to have trivial arcs. An earlier description of our map π ↦ → πˆ followed a suggestion by Lin [13] and used growth diagrams à la Krattenthaler [12] (as in work by Kasraoui [9] and by Yan [20]). It is straightforward to confirm that π ↦ → πˆ is a bijection from partitions of subsets of [n] to partitions of [n + 1], and that it increases the distance of k-crossings (and k-nestings) by one. In particular, (δ + 1)-distant k-crossings in πˆ correspond to δ -distant k-crossings in π , which for δ ∈ {0, 1} correspond to δ -distant k-crossings in the standardization std(π ).2 However, for δ ≥ 2, the distance of a k-crossing in π may not be the same as in std(π ). For example, the 3-distant crossing 1 < 2 < 5 < 6 in πˆ = {15, 26, 3, 4} is the image of the 2-distant crossing 1 < 2 < 4 < 5 in π = {14, 25}, which corresponds to the 1-distant crossing 1 < 2 < 3 < 4 in std(π ) = {13, 24}. This explains why NCδ+1,k (n + 1) is not the binomial transform of the sequence NCδ,k (n) for δ ≥ 2. The following lemma follows immediately from our definitions. Lemma 5. The extension map π ↦ → πˆ satisfies the following properties, where S is any sequence (a1 , . . . , ak , b1 , . . . , bk ), ˆ S = (a1 , . . . , ak , b1 + 1, . . . , bk + 1), δ ≥ 0, and k ≥ 1. (i) The sequence S is a k-crossing of distance δ in π if and only if ˆ S is a k-crossing of distance δ + 1 in πˆ . (ii) The sequence S is a δ -distant k-crossing after the relabeling from π to std(π ) only if ˆ S is a (δ + 1)-distant k-crossing in πˆ . (iii) The sequence S is an enhanced (resp. classical) k-crossing after the relabeling from π to std(π ) if and only if ˆ S is a classical (resp. 2-distant) k-crossing in πˆ . Moreover, analogous properties hold for nestings. 2 If π is a partition of X ⊆ [n], then std(π ) is obtained by relabeling the elements of π to 1, 2, . . . , |X |, while preserving their order. Please cite this article as: J.B. Gil and J.O. Tirrell, A simple bijection for enhanced, classical, and 2-distant k-noncrossing partitions, Discrete Mathematics (2019) 111705, https://doi.org/10.1016/j.disc.2019.111705.
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J.B. Gil and J.O. Tirrell / Discrete Mathematics xxx (xxxx) xxx Table 2 Values of dcrδ,k and dneδ,k for π = {1479, 25, 6}. dcr0,1
dcr1,1
dcr2,1
dcr0,2
dcr1,2
dcr2,2
dcr0,3
dcr1,3
dcr2,3
5 5 7
4 4 5
3 4 4
4 4 6
2 2 4
0 1 2
1 1 1
0 0 1
0 0 0
dne0,1
dne1,1
dne2,1
dne0,2
dne1,2
dne2,2
dne0,3
dne1,3
dne2,3
5 5 7
4 4 5
3 4 4
1 1 4
0 0 1
0 0 0
0 0 0
0 0 0
0 0 0
std(π )
π πˆ
std(π )
π πˆ
In particular, if dcrδ,k (π ) (resp. dneδ,k (π )) denotes the number of δ -distant k-crossings (resp. k-nestings) of a given partition π , then dcrδ,k (std(π )) ≤ dcrδ,k (π ) = dcrδ+1,k (πˆ ), and
(5)
dneδ,k (std(π )) ≤ dneδ,k (π ) = dneδ+1,k (πˆ ) for all δ ≥ 0, k ≥ 1,
(6)
with equality when δ ∈ {0, 1}. See Table 2 for examples. Lemma 5 immediately gives us the following generalization of Theorems 1 and 3. Theorem 6. For any n ≥ 0 and given sequences α, α ′ , β , and β ′ , let Pδ (n, α, α ′ , β, β ′ ) denote the number of partitions of [n] with dcrδ,k = αk , dcrδ+1,k = αk′ , dneδ,k = βk , dneδ+1,k = βk′ for all k ≥ 1. Then P1 (n + 1, α, α ′ , β, β ′ ) =
n ( ) ∑ n i=0
i
P0 (i, α, α ′ , β, β ′ ).
(7)
Summing over all α, α ′ , β, β ′ with either αk = 0 or αk′ = 0 gives Eq. (1) or (2) of Theorem 1. Summing over all α, α ′ , β, β ′ with either αk = 0 or αk′ = 0, and either βj = 0 or βj′ = 0 gives the different δ, ε ∈ {0, 1} cases of Theorem 3. Summing over all α, α ′ , β, β ′ with some fixed α1 , and α2 = 0, we obtain a result equivalent to [4, Theorem 3.2]. In fact, the ‘‘reduction algorithm" used in [4] is essentially πˆ ↦ → π , the inverse of our extension map. Remark 7. Note that placing restrictions on dcrδ,k and dneδ,k for δ ≥ 0 and k ≥ 1 allows us to control many important properties of partitions. Below we highlight a few key features. Similar statements hold for nestings.
▷ dcrδ,k (π ) − dcrδ+1,k (π ) is the number of k-crossings of distance exactly δ . ▷ dcr0,1 (π ) − dcr1,1 (π ) = dne0,1 (π ) − dne1,1 (π ) is simply the number of singleton blocks, whose elements are both minimal and maximal in their block.
▷ dcr0,2 (π ) − dcr1,2 (π ) is the number of transients, elements which are neither minimal nor maximal in their block. ▷ dcr1,1 (π ) = dne1,1 (π ) is the number of edges in the arc diagram of π , which equals n − k when π is a partition of [n] with k blocks. ▷ dcrδ,k (π ) = 0 if and only if π is δ -distant k-noncrossing. ▷ dcrm,1 (π ) − dcr1,1 (π ) = dnem,1 (π ) − dne1,1 (π ) is the number of consecutive pairs in blocks at distance less than m. When this is zero, π is said to be m-regular. 3. Further remarks 3.1. Partitions of [n] to partitions of [n + 1] If we restrict our map π ↦ → πˆ to partitions of [n] such that std(π ) = π (i.e. there is no i ∈ [n] which is not contained in some block of π ), we obtain a bijection between partitions π of [n] and partitions πˆ of [n + 1] having no i ∈ [n] such that i is maximal within its block in πˆ and i + 1 is minimal within its block in πˆ . Alternatively, we can map a partition π of [n] into a partition of [n + 1] by first removing its singletons (denote the modified singleton free partition by π ′ ), and then applying the extension map to π ′ to obtain a partition πˆ ′ of [n + 1]. For example, if π = {1478, 25, 3, 6}, then π ′ = {1478, 25} and πˆ ′ = {15, 26, 3, 48, 79}. Note that singletons only occur in enhanced k-nestings where the distance is zero, not in classical k-nestings. Moreover, singletons occur in enhanced k-crossings only when k = 1 and the distance is zero. Therefore, dneδ,k (π ) = dneδ,k (π ′ ) = dneδ+1,k (πˆ ′ ) for all δ, k ≥ 1.
(8)
And for all δ ≥ 0, k ≥ 1, except the case δ = 0 and k = 1, dcrδ,k (π ) = dcrδ,k (π ′ ) = dcrδ+1,k (πˆ ′ ).
(9)
Please cite this article as: J.B. Gil and J.O. Tirrell, A simple bijection for enhanced, classical, and 2-distant k-noncrossing partitions, Discrete Mathematics (2019) 111705, https://doi.org/10.1016/j.disc.2019.111705.
J.B. Gil and J.O. Tirrell / Discrete Mathematics xxx (xxxx) xxx
5
This gives the following. Theorem 8. For n ≥ 0, and for (δ ≥ 0, k > 1) or (δ > 0, k ≥ 1), let Πδ,k (n, α, β ) be the set of partitions of [n] with dcrδ,k = α and dneδ,k = β . The map π ↦ → πˆ ′ gives a bijection between Πδ,k (n, α, β ) and the set of partitions of [n + 1] with dcr1,1 = dcr1,2 (i.e. 2-regular), dcrδ+1,k = α , and dneδ+1,k = β . 3.2. Partitions without singleton blocks Let NCδ,k (n) denote the number of δ -distant k-noncrossing partitions of [n] with no singleton blocks. Then NCδ,k (n) =
n ( ) ∑ n i=0
i
NCδ,k (i),
and it can be easily checked that NCδ,k (n + 1) =
n ( )( ∑ n i=0
i
)
NCδ,k (i) + NCδ,k (i + 1) .
Using the inverse binomial transform, we then get that Theorem 1 is equivalent to NC0,k (n) = NC1,k (n) + NC1,k (n + 1), and NC1,k (n) = NC2,k (n) + NC2,k (n + 1). From these equations, one obtains that NC1,2 (n) gives the sequence of Riordan numbers A005043, and that NC2,2 (n) gives the sequence A033297. The sequence NC0,3 (n) has been shown by Bostan, the second author, Westbury, and Zhang [1] to be the sequence A059710, which appears in the representation theory of G2 (see [19]). Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] A. Bostan, J. Tirrell, B.W. Westbury, Y. Zhang, On sequences associated with the invariant theory of rank two simple Lie algebras, in preparation. [2] M. Bousquet-Mélou, G. Xin, On partitions avoiding 3-crossings, Sém. Lothar. Combin. 54 (2006) Article B54e. [3] S. Burrill, S. Elizalde, M. Mishna, L. Yen, A generating tree approach to k-nonnesting partitions and permutations, Ann. Combin. 20 (3) (2016) 453–485. [4] W.Y.C. Chen, E.Y.P. Deng, R.R.X. Du, Reduction of m-regular noncrossing partitions, European J. Combin. 26 (2005) 237–243. [5] W.Y.C. Chen, E.Y.P. Deng, R.R.X. Du, R.P. Stanley, C.H. Yan, Crossings and nestings of matchings and partitions, Trans. Amer. Math. Soc. 359 (4) (2007) 1555–1575. [6] D. Drake, J.S. Kim, k-distant crossings and nestings of matchings and partitions, in: DMTCS Proceedings, AK, 2009, pp. 349–360. [7] I.M. Gessel, J.S. Kim, A note on 2-distant noncrossing partitions and weighted Motzkin paths, Discrete Math. 310 (23) (2010) 3421–3425. [8] E.Y. Jin, J. Qin, C.M. Reidys, Combinatorics of RNA structures with pseudoknots, Bull. Math. Biol. 70 (2008) 45–67. [9] A. Kasraoui, d-Regular set partitions and rook placements, Sém. Lothar. Combin. 62 (2009) Article B62a. [10] J.S. Kim, Bijections on two variations of noncrossing partitions, Discrete Math. 311 (2011) 1057–1063. [11] J.S. Kim, Front representations of set partitions, SIAM J. Discrete Math. 25 (1) (2011) 447–461. [12] C. Krattenthaler, Growth diagrams, and increasing and decreasing chains in fillings of Ferrers shapes, Adv. Appl. Math. 37 (3) (2006) 404–431. [13] Z. Lin, Restricted inversion sequences and enhanced 3-noncrossing partitions, European J. Combin. 70 (2018) 202–211. [14] Z. Lin, D. Kim, A combinatorial bijection on k-noncrossing partitions, arXiv:1905.10526. [15] E. Marberg, Crossings and nestings in colored set partitions, Electron. J. Combin. 20 (4) (2013) #P6. [16] M. Mishna, L. Yen, Set partitions with no m-nesting, in: Advances in Combinatorics, Springer, Heidelberg, 2013, pp. 249–258. [17] OEIS Foundation Inc., The on-line encyclopedia of integer sequences, 2018. [18] R.P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge University Press, Cambridge, 1999. [19] B.W. Westbury, Enumeration of non-positive planar trivalent graphs, J. Algebraic Combin. 25 (2007) 357–373. [20] S.H.F. Yan, Bijections for inversion sequences, ascent sequences and 3-nonnesting set partitions, Appl. Math. Comput. 325 (2018) 24–30.
Please cite this article as: J.B. Gil and J.O. Tirrell, A simple bijection for enhanced, classical, and 2-distant k-noncrossing partitions, Discrete Mathematics (2019) 111705, https://doi.org/10.1016/j.disc.2019.111705.
Contents lists available at ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
A simple bijection for enhanced, classical, and 2-distant k-noncrossing partitions✩ ∗
Juan B. Gil a , , Jordan O. Tirrell b a b
Penn State Altoona, 3000 Ivyside Park, Altoona, PA 16601, USA Department of Mathematics and Computer Science, Washington College, Chestertown, MD 21620, USA
article
info
a b s t r a c t In this note, we give a simple extension map from partitions of subsets of [n] to partitions of [n + 1], which sends δ -distant k-crossings to (δ + 1)-distant k-crossings (and similarly for nestings). This map provides a combinatorial proof of the fact that the numbers of enhanced, classical, and 2-distant k-noncrossing partitions are each related to the next via the binomial transform. Our work resolves a recent conjecture of Zhicong Lin and generalizes earlier reduction identities for partitions. © 2019 Elsevier B.V. All rights reserved.
Article history: Received 3 December 2018 Received in revised form 4 August 2019 Accepted 17 October 2019 Available online xxxx Keywords: k-noncrossing partition Enhanced k-noncrossing partition
1. Introduction Given a partition π of a set of integers and k ≥ 1, the arc diagram of π is obtained by drawing an arc between each pair of integers that appears consecutively in the same block of π . For example, the partition π = {148, 26, 3, 57} can be represented as
1
2
3
4
5
6
7
8
and we say that 1246, 2468 and 2567 are crossings, and 4578 is a nesting. Additionally, we say 148 is an enhanced crossing and 134 is an enhanced nesting. Precisely, an enhanced k-crossing is a sequence a1 < a2 < · · · < ak ≤ b1 < b2 < · · · < bk such that there is an arc between each pair ai , bi , and we consider singleton blocks to have trivial arcs. A classical k-crossing additionally requires that ak < b1 , and in general a δ -distant k-crossing requires that the distance b1 − ak is at least δ . Thus, the enhanced crossings correspond to δ = 0 and the classical crossings correspond to δ = 1. Note that an enhanced 1-crossing is simply an arc, and a classical 1-crossing is a nontrivial arc. An enhanced/classical/δ -distant k-nesting is defined similarly with an arc between each pair ai , bk+1−i . Returning to our example above, there is one enhanced ✩ The authors would like to thank the Department of Mathematics and Statistics and the Hutchcroft fund at Mount Holyoke College for their support. ∗ Corresponding author. E-mail addresses: [email protected] (J.B. Gil), [email protected] (J.O. Tirrell). https://doi.org/10.1016/j.disc.2019.111705 0012-365X/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: J.B. Gil and J.O. Tirrell, A simple bijection for enhanced, classical, and 2-distant k-noncrossing partitions, Discrete Mathematics (2019) 111705, https://doi.org/10.1016/j.disc.2019.111705.
2
J.B. Gil and J.O. Tirrell / Discrete Mathematics xxx (xxxx) xxx Table 1 Sequences NCδ,k (n) for small k and δ = 0, 1, 2. n
0
1
2
3
4
5
6
7
8
9
10
11
OEIS[17]
NC0,1 (n) NC0,2 (n) NC0,3 (n) NC0,4 (n) NC0,5 (n)
1 1 1 1 1
0 1 1 1 1
0 2 2 2 2
0 4 5 5 5
0 9 15 15 15
0 21 51 52 52
0 51 191 203 203
0 127 772 876 877
0 323 3320 4120 4140
0 835 15032 20883 21146
0 2188 71084 113034 115945
0 5798 348889 648410 678012
A001006 A108307 A192855 A192865
NC1,1 (n) NC1,2 (n) NC1,3 (n) NC1,4 (n) NC1,5 (n)
1 1 1 1 1
1 1 1 1 1
1 2 2 2 2
1 5 5 5 5
1 14 15 15 15
1 42 52 52 52
1 132 202 203 203
1 429 859 877 877
1 1430 3930 4139 4140
1 4862 19095 21119 21147
1 16796 97566 115495 115974
1 58786 520257 671969 678530
A000108 A108304 A108305 A192126
NC2,1 (n) NC2,2 (n) NC2,3 (n) NC2,4 (n) NC2,5 (n)
1 1 1 1 1
1 1 1 1 1
2 2 2 2 2
4 5 5 5 5
8 15 15 15 15
16 51 52 52 52
32 188 203 203 203
64 731 876 877 877
128 2950 4115 4140 4140
256 12235 20765 21146 21147
512 51822 111301 115938 115975
1024 223191 627821 677765 678569
A000079 A007317 A318191 A318192 A318193
B(n)
1
1
2
5
15
52
203
877
4140
21147
115975
678570
A000110
3-crossing 12468, and no enhanced 3-nesting. Of the classical crossings and nestings, all but one are 2-distant, and none are 3-distant. Much of the combinatorial interest in these structures follows the work of Chen, Deng, Du, Stanley, and Yan [5]. They proved a beautiful bijective symmetry between k-crossings and k-nestings, which D. Drake & J.S. Kim [6] generalized to δ -distant k-crossings and k-nestings (see Eq. (3)). Motivation for δ -distant k-crossings also comes from the study of RNA structures [8]. Let NCδ,k (n) denote the number of δ -distant k-noncrossing partitions of [n] = {1, . . . , n}, that is, partitions of [n] with no δ -distant k-crossing. Examples are given in Table 1. In this paper, we give combinatorial proofs of the following binomial transform identities. Theorem 1.
For integers n ≥ 0 and k ≥ 1,
NC1,k (n + 1) =
n ( ) ∑ n i=0 n
NC2,k (n + 1) =
i
NC0,k (i) and
∑ ( n) i=0
i
NC1,k (i).
(1)
(2)
Eq. (1) is well-known for k = 2, and it was recently proven for k = 3 by Lin [13], who conjectured it held for all k. In the final version of [13], Lin announced two different combinatorial proofs of his conjecture in( joint ) work with D. Kim [14]. 2n 1 Eq. (2) appears to be new for k > 2. For k = 2, it is well-known that NC1,2 (n) = n+ . The fact that NC2,2 (n) is 1 n the binomial transform of NC1,2 (n) was shown by Drake & J.S. Kim [6] using Charlier diagrams and generating functions. Several bijections with other objects can be found in [7,10]. Bousquet-Mélou & Xin [2] showed that the sequences NC0,3 (n) and NC1,3 (n) are P-recursive and gave explicit recurrences. Mishna & Yen [16] used generating trees to find functional equations for NC1,k (n) when k > 3. Burrill, Elizalde, Mishna, and Yen [3] did the same for NC0,k (n). Theorem 1 provides a simple way to connect the enumerations of NC0,k (n), NC1,k (n), and NC2,k (n). In particular, as Lin [13] observed, the binomial transform preserves D-finiteness (see [18, Theorem 6.4.10]). Corollary 2. The D-finiteness of the ordinary generating functions for NC0,k (n), NC1,k (n), and NC2,k (n) are the same.1 Unfortunately, the sequence NC3,k (n + 1) is not the binomial transform of the sequence NC2,k (n), and in general, not much is known about NCδ,k (n) for δ ≥ 3 (the generating function for NC3,2 (n) is given in [11]). Let NCNδ,k,ε,j (n) denote the number of δ -distant k-noncrossing partitions of [n] which also contain no ε -distant j-nesting. By the celebrated symmetry of Chen et al. [5], generalized by Drake & Kim [6], we have NCNδ,k,ε,j (n) = NCNε,j,δ,k (n).
(3)
We will give a combinatorial proof of the following refinement of Theorem 1. 1 For k > 3, it is conjectured [2] that the generating function of NC (n) is not D-finite. 1 ,k Please cite this article as: J.B. Gil and J.O. Tirrell, A simple bijection for enhanced, classical, and 2-distant k-noncrossing partitions, Discrete Mathematics (2019) 111705, https://doi.org/10.1016/j.disc.2019.111705.
J.B. Gil and J.O. Tirrell / Discrete Mathematics xxx (xxxx) xxx
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Fig. 1. The arc diagram for π = {1479, 25, 6} extends to πˆ = {15, 267T, 3, 48, 9}.
Theorem 3.
For integers n ≥ 0, j, k ≥ 1, and ε, δ ∈ {0, 1},
NCNδ+1,k,ε+1,j (n + 1) =
n ( ) ∑ n i=0
i
NCNδ,k,ε,j (i)
(4)
The generating function for NCN1,k,1,j (n) is not simply D-finite but in fact rational for all k, j ≥ 1. This was shown by Marberg [15] using a bijection with walks on certain multigraphs (and he actually proved this in the more general setting of colored partitions). By Theorem 3, the generating functions for NCN0,k,0,j (n) and NCN2,k,2,j (n) are also rational. 2. Main bijection It is known that the sequence B(n) of Bell numbers, counts the number of partitions of [n], is an eigensequence ) ∑n (which n of the binomial transform. That is, B(n + 1) = i=0 i B(i). The usual combinatorial proof of this fact uses a bijection from partitions of subsets of [n] to partitions of [n + 1], defined by adding a new block containing n + 1 together with every element of [n] not already contained in a block. For example, if n = 9, this bijection takes {1479, 25, 6} to {1479, 25, 6, 38T}, where T stands for 10. We now give a different bijection which behaves well with respect to crossings and distance. Definition 4. For any n ≥ 0, and for any partition π of some subset of [n], we define the partition πˆ of [n + 1] as follows. For each pair of vertices v < w appearing consecutively in the same block of π , place v and w + 1 in the same block of πˆ . For each singleton u of π , place u and u + 1 in the same block of πˆ . At last, place any remaining vertices in [n + 1] as singletons of πˆ . We will refer to π ↦ → πˆ as the extension map. For example, if n = 9 and π = {1479, 25, 6}, then πˆ = {15, 267T, 3, 48, 9}. An elegant geometric description of this map can be obtained by extending the lines in the arc diagram of π to arrive at the arc diagram of πˆ (see Fig. 1), where singletons are considered to have trivial arcs. An earlier description of our map π ↦ → πˆ followed a suggestion by Lin [13] and used growth diagrams à la Krattenthaler [12] (as in work by Kasraoui [9] and by Yan [20]). It is straightforward to confirm that π ↦ → πˆ is a bijection from partitions of subsets of [n] to partitions of [n + 1], and that it increases the distance of k-crossings (and k-nestings) by one. In particular, (δ + 1)-distant k-crossings in πˆ correspond to δ -distant k-crossings in π , which for δ ∈ {0, 1} correspond to δ -distant k-crossings in the standardization std(π ).2 However, for δ ≥ 2, the distance of a k-crossing in π may not be the same as in std(π ). For example, the 3-distant crossing 1 < 2 < 5 < 6 in πˆ = {15, 26, 3, 4} is the image of the 2-distant crossing 1 < 2 < 4 < 5 in π = {14, 25}, which corresponds to the 1-distant crossing 1 < 2 < 3 < 4 in std(π ) = {13, 24}. This explains why NCδ+1,k (n + 1) is not the binomial transform of the sequence NCδ,k (n) for δ ≥ 2. The following lemma follows immediately from our definitions. Lemma 5. The extension map π ↦ → πˆ satisfies the following properties, where S is any sequence (a1 , . . . , ak , b1 , . . . , bk ), ˆ S = (a1 , . . . , ak , b1 + 1, . . . , bk + 1), δ ≥ 0, and k ≥ 1. (i) The sequence S is a k-crossing of distance δ in π if and only if ˆ S is a k-crossing of distance δ + 1 in πˆ . (ii) The sequence S is a δ -distant k-crossing after the relabeling from π to std(π ) only if ˆ S is a (δ + 1)-distant k-crossing in πˆ . (iii) The sequence S is an enhanced (resp. classical) k-crossing after the relabeling from π to std(π ) if and only if ˆ S is a classical (resp. 2-distant) k-crossing in πˆ . Moreover, analogous properties hold for nestings. 2 If π is a partition of X ⊆ [n], then std(π ) is obtained by relabeling the elements of π to 1, 2, . . . , |X |, while preserving their order. Please cite this article as: J.B. Gil and J.O. Tirrell, A simple bijection for enhanced, classical, and 2-distant k-noncrossing partitions, Discrete Mathematics (2019) 111705, https://doi.org/10.1016/j.disc.2019.111705.
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J.B. Gil and J.O. Tirrell / Discrete Mathematics xxx (xxxx) xxx Table 2 Values of dcrδ,k and dneδ,k for π = {1479, 25, 6}. dcr0,1
dcr1,1
dcr2,1
dcr0,2
dcr1,2
dcr2,2
dcr0,3
dcr1,3
dcr2,3
5 5 7
4 4 5
3 4 4
4 4 6
2 2 4
0 1 2
1 1 1
0 0 1
0 0 0
dne0,1
dne1,1
dne2,1
dne0,2
dne1,2
dne2,2
dne0,3
dne1,3
dne2,3
5 5 7
4 4 5
3 4 4
1 1 4
0 0 1
0 0 0
0 0 0
0 0 0
0 0 0
std(π )
π πˆ
std(π )
π πˆ
In particular, if dcrδ,k (π ) (resp. dneδ,k (π )) denotes the number of δ -distant k-crossings (resp. k-nestings) of a given partition π , then dcrδ,k (std(π )) ≤ dcrδ,k (π ) = dcrδ+1,k (πˆ ), and
(5)
dneδ,k (std(π )) ≤ dneδ,k (π ) = dneδ+1,k (πˆ ) for all δ ≥ 0, k ≥ 1,
(6)
with equality when δ ∈ {0, 1}. See Table 2 for examples. Lemma 5 immediately gives us the following generalization of Theorems 1 and 3. Theorem 6. For any n ≥ 0 and given sequences α, α ′ , β , and β ′ , let Pδ (n, α, α ′ , β, β ′ ) denote the number of partitions of [n] with dcrδ,k = αk , dcrδ+1,k = αk′ , dneδ,k = βk , dneδ+1,k = βk′ for all k ≥ 1. Then P1 (n + 1, α, α ′ , β, β ′ ) =
n ( ) ∑ n i=0
i
P0 (i, α, α ′ , β, β ′ ).
(7)
Summing over all α, α ′ , β, β ′ with either αk = 0 or αk′ = 0 gives Eq. (1) or (2) of Theorem 1. Summing over all α, α ′ , β, β ′ with either αk = 0 or αk′ = 0, and either βj = 0 or βj′ = 0 gives the different δ, ε ∈ {0, 1} cases of Theorem 3. Summing over all α, α ′ , β, β ′ with some fixed α1 , and α2 = 0, we obtain a result equivalent to [4, Theorem 3.2]. In fact, the ‘‘reduction algorithm" used in [4] is essentially πˆ ↦ → π , the inverse of our extension map. Remark 7. Note that placing restrictions on dcrδ,k and dneδ,k for δ ≥ 0 and k ≥ 1 allows us to control many important properties of partitions. Below we highlight a few key features. Similar statements hold for nestings.
▷ dcrδ,k (π ) − dcrδ+1,k (π ) is the number of k-crossings of distance exactly δ . ▷ dcr0,1 (π ) − dcr1,1 (π ) = dne0,1 (π ) − dne1,1 (π ) is simply the number of singleton blocks, whose elements are both minimal and maximal in their block.
▷ dcr0,2 (π ) − dcr1,2 (π ) is the number of transients, elements which are neither minimal nor maximal in their block. ▷ dcr1,1 (π ) = dne1,1 (π ) is the number of edges in the arc diagram of π , which equals n − k when π is a partition of [n] with k blocks. ▷ dcrδ,k (π ) = 0 if and only if π is δ -distant k-noncrossing. ▷ dcrm,1 (π ) − dcr1,1 (π ) = dnem,1 (π ) − dne1,1 (π ) is the number of consecutive pairs in blocks at distance less than m. When this is zero, π is said to be m-regular. 3. Further remarks 3.1. Partitions of [n] to partitions of [n + 1] If we restrict our map π ↦ → πˆ to partitions of [n] such that std(π ) = π (i.e. there is no i ∈ [n] which is not contained in some block of π ), we obtain a bijection between partitions π of [n] and partitions πˆ of [n + 1] having no i ∈ [n] such that i is maximal within its block in πˆ and i + 1 is minimal within its block in πˆ . Alternatively, we can map a partition π of [n] into a partition of [n + 1] by first removing its singletons (denote the modified singleton free partition by π ′ ), and then applying the extension map to π ′ to obtain a partition πˆ ′ of [n + 1]. For example, if π = {1478, 25, 3, 6}, then π ′ = {1478, 25} and πˆ ′ = {15, 26, 3, 48, 79}. Note that singletons only occur in enhanced k-nestings where the distance is zero, not in classical k-nestings. Moreover, singletons occur in enhanced k-crossings only when k = 1 and the distance is zero. Therefore, dneδ,k (π ) = dneδ,k (π ′ ) = dneδ+1,k (πˆ ′ ) for all δ, k ≥ 1.
(8)
And for all δ ≥ 0, k ≥ 1, except the case δ = 0 and k = 1, dcrδ,k (π ) = dcrδ,k (π ′ ) = dcrδ+1,k (πˆ ′ ).
(9)
Please cite this article as: J.B. Gil and J.O. Tirrell, A simple bijection for enhanced, classical, and 2-distant k-noncrossing partitions, Discrete Mathematics (2019) 111705, https://doi.org/10.1016/j.disc.2019.111705.
J.B. Gil and J.O. Tirrell / Discrete Mathematics xxx (xxxx) xxx
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This gives the following. Theorem 8. For n ≥ 0, and for (δ ≥ 0, k > 1) or (δ > 0, k ≥ 1), let Πδ,k (n, α, β ) be the set of partitions of [n] with dcrδ,k = α and dneδ,k = β . The map π ↦ → πˆ ′ gives a bijection between Πδ,k (n, α, β ) and the set of partitions of [n + 1] with dcr1,1 = dcr1,2 (i.e. 2-regular), dcrδ+1,k = α , and dneδ+1,k = β . 3.2. Partitions without singleton blocks Let NCδ,k (n) denote the number of δ -distant k-noncrossing partitions of [n] with no singleton blocks. Then NCδ,k (n) =
n ( ) ∑ n i=0
i
NCδ,k (i),
and it can be easily checked that NCδ,k (n + 1) =
n ( )( ∑ n i=0
i
)
NCδ,k (i) + NCδ,k (i + 1) .
Using the inverse binomial transform, we then get that Theorem 1 is equivalent to NC0,k (n) = NC1,k (n) + NC1,k (n + 1), and NC1,k (n) = NC2,k (n) + NC2,k (n + 1). From these equations, one obtains that NC1,2 (n) gives the sequence of Riordan numbers A005043, and that NC2,2 (n) gives the sequence A033297. The sequence NC0,3 (n) has been shown by Bostan, the second author, Westbury, and Zhang [1] to be the sequence A059710, which appears in the representation theory of G2 (see [19]). Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] A. Bostan, J. Tirrell, B.W. Westbury, Y. Zhang, On sequences associated with the invariant theory of rank two simple Lie algebras, in preparation. [2] M. Bousquet-Mélou, G. Xin, On partitions avoiding 3-crossings, Sém. Lothar. Combin. 54 (2006) Article B54e. [3] S. Burrill, S. Elizalde, M. Mishna, L. Yen, A generating tree approach to k-nonnesting partitions and permutations, Ann. Combin. 20 (3) (2016) 453–485. [4] W.Y.C. Chen, E.Y.P. Deng, R.R.X. Du, Reduction of m-regular noncrossing partitions, European J. Combin. 26 (2005) 237–243. [5] W.Y.C. Chen, E.Y.P. Deng, R.R.X. Du, R.P. Stanley, C.H. Yan, Crossings and nestings of matchings and partitions, Trans. Amer. Math. Soc. 359 (4) (2007) 1555–1575. [6] D. Drake, J.S. Kim, k-distant crossings and nestings of matchings and partitions, in: DMTCS Proceedings, AK, 2009, pp. 349–360. [7] I.M. Gessel, J.S. Kim, A note on 2-distant noncrossing partitions and weighted Motzkin paths, Discrete Math. 310 (23) (2010) 3421–3425. [8] E.Y. Jin, J. Qin, C.M. Reidys, Combinatorics of RNA structures with pseudoknots, Bull. Math. Biol. 70 (2008) 45–67. [9] A. Kasraoui, d-Regular set partitions and rook placements, Sém. Lothar. Combin. 62 (2009) Article B62a. [10] J.S. Kim, Bijections on two variations of noncrossing partitions, Discrete Math. 311 (2011) 1057–1063. [11] J.S. Kim, Front representations of set partitions, SIAM J. Discrete Math. 25 (1) (2011) 447–461. [12] C. Krattenthaler, Growth diagrams, and increasing and decreasing chains in fillings of Ferrers shapes, Adv. Appl. Math. 37 (3) (2006) 404–431. [13] Z. Lin, Restricted inversion sequences and enhanced 3-noncrossing partitions, European J. Combin. 70 (2018) 202–211. [14] Z. Lin, D. Kim, A combinatorial bijection on k-noncrossing partitions, arXiv:1905.10526. [15] E. Marberg, Crossings and nestings in colored set partitions, Electron. J. Combin. 20 (4) (2013) #P6. [16] M. Mishna, L. Yen, Set partitions with no m-nesting, in: Advances in Combinatorics, Springer, Heidelberg, 2013, pp. 249–258. [17] OEIS Foundation Inc., The on-line encyclopedia of integer sequences, 2018. [18] R.P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge University Press, Cambridge, 1999. [19] B.W. Westbury, Enumeration of non-positive planar trivalent graphs, J. Algebraic Combin. 25 (2007) 357–373. [20] S.H.F. Yan, Bijections for inversion sequences, ascent sequences and 3-nonnesting set partitions, Appl. Math. Comput. 325 (2018) 24–30.
Please cite this article as: J.B. Gil and J.O. Tirrell, A simple bijection for enhanced, classical, and 2-distant k-noncrossing partitions, Discrete Mathematics (2019) 111705, https://doi.org/10.1016/j.disc.2019.111705.