- Email: [email protected]
Enumeration Problems on the Expansion of a Chord Diagram
Available online at www.sciencedirect.com
Electronic Notes in Discrete Mathematics 54 (2016) 51–56
www.elsevier.com/locate/endm
Enumeration Problems on the Expansion of a Chord Diagram Tomoki Nakamigawa
1,2
Department of Information Science Shonan Institute of Technology 1-1-25 Tsujido-Nishikaigan, Fujisawa, Kanagawa 251-8511, Japan
Abstract A chord diagram is a set of chords of a circle such that no pair of chords has a common endvertex. A pair of chords is called a crossing if the two chords intersect. A chord diagram E is called nonintersecting if E contains no crossing. For a chord diagram E having a crossing S = {x1 x3 , x2 x4 }, the expansion of E with respect to S is to replace E with E1 = (E \ S) ∪ {x2 x3 , x4 x1 } or E2 = (E \ S) ∪ {x1 x2 , x3 x4 }. A chord diagram E = E1 ∪ E2 is called complete bipartite of type (m, n), denoted by Cm,n , if (1) both E1 and E2 are nonintersecting, (2) for every pair e1 ∈ E1 and e2 ∈ E2 , e1 and e2 are crossing, and (3) |E1 | = m, |E2 | = n. Let fm,n be the cardinality of the multiset of all nonintersecting chord diagrams generated from Cm,n with a finite sequence of expansions. In this paper, it is shown m,n fm,n (xm /m!)(y n /n!) is 1/(cosh x cosh y − (sinh x + sinh y)). Keywords: enumeration, chord diagram, expansion, alternating permutation
1 2
This work was supported by KAKENHI(16K05260). Email:[email protected]
http://dx.doi.org/10.1016/j.endm.2016.09.010 1571-0653/© 2016 Elsevier B.V. All rights reserved.
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T. Nakamigawa / Electronic Notes in Discrete Mathematics 54 (2016) 51–56
Fig. 1. The expansion of a chord diagram.
1
Introduction
Let us consider a set of chords of a circle. A set of chords is called a chord diagram, if they have no common endvertex. If a chord diagram consists of a set of n mutually crossing chords, it is called an n-crossing. A 2-crossing is simply called a crossing as well. If a chord diagram contains no crossing, it is called nonintersecting. Let V be a set of 2n vertices on a circle, and let E be a chord diagram of order n, where each chord has endvertices of V . We denote the family of all such chord diagrams by CD(V ). Let x1 , x2 , x3 , x4 ∈ V be placed on a circle in clockwise order. Let E ∈ CD(V ). For a crossing S = {x1 x3 , x2 x4 } ⊂ E, let S1 = {x2 x3 , x4 x1 }, and S2 = {x1 x2 , x3 x4 }. The expansion of E with respect to S is defined as a replacement of E with E1 = (E \ S) ∪ S1 or E2 = (E \ S) ∪ S2 (see Figure 1). For a chord diagram E ∈ CD(V ) and a crossing S ⊂ E, we have two successors E1 and E2 of E. By iterating possible expansions, we have a multiset of nonintersecting chord diagrams. As is remarked in [1], the resulting multiset of nonintersecting chord diagrams generated from E with a set of expansions is uniquely determined. Let us denote the multiset of nonintersecting chord diagrams generated by E ∈ CD(V ) by N CD(E). For E ∈ CD(V ), let us define f (E) as the cardinality of N CD(E) as a multiset. Let Cn be an n-crossing. In [1], the exponential generating function of
T. Nakamigawa / Electronic Notes in Discrete Mathematics 54 (2016) 51–56
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Fig. 2. Complete bipartite chord diagram Cm,n , where m = 4 and n = 3.
f (Cn ) is shown as 1 xn f (Cn ) = n! 1 − sin x n≥0 x3 x4 x5 x6 x2 + 5 + 16 + 61 + 272 + · · · . 2! 3! 4! 5! 6! Hence, f (Cn ) = En+1 , where En is called the Euler number, which corresponds to the number of alternating permutations of order n (see [2] for an excellent survey of alternating permutations). A chord diagram E is called complete bipartite of type (m, n), where m and n are nonnegative integers, if E is paratitioned into E1 and E2 such that (1) both E1 and E2 are nonintersecting, (2) for every pair of chords e1 ∈ E1 and e2 ∈ E2 , e1 and e2 are crossing, and (3) |E1 | = m, |E2 | = n. (See Fig.2.) Let us denote a complete bipartite chord diagram of type (m, n) by Cm,n , and let us denote f (Cm,n ) by fm,n . In this paper, we focus on counting fm,n . =1 + x + 2
2
Main Results
Let define the exponential generating function F (x, y) as us m n m≥0 n≥0 fm,n (x /m!)(y /n!). The main result of the paper is the following theorem. Theorem 2.1 F (x, y) =
1 . cosh x cosh y − (sinh x + sinh y)
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T. Nakamigawa / Electronic Notes in Discrete Mathematics 54 (2016) 51–56
Fig. 3. Am,n,k (left) and Bm,n,k (right), where m = 4, n = 3 and k = 2.
3
Proof of Theorem 2.1
Let us denote the righthandside of the equation of Theorem 2.1 by g(x, y). Then we have g(x, y) = g(y, x), g(x, 0) = ex and ∂ ∂g = (g · sinh y). ∂x ∂y On the other hand, it is easy to see that F (x, y) = F (y, x) and F (x, 0) = ex . It is left for us to check that F (x, y) also satisfies that ∂ ∂F = (F · sinh y). ∂x ∂y Firstly, note that the equation (1) is equivalent to the following recurrence. n+1 (2) fm+1,n = fm,n+1−i i i:odd,1≤i≤n+1 (1)
for all m ≥ 0 and n ≥ 0. In the following, we will show the recurrence (2). Let us introduce new families of chord diagrams. Let E(m, n) = E1 ∪E2 be isomorphic to a complete bipartite chord diagram of type (m, n) with |E1 | = m and |E2 | = n. For 0 ≤ k ≤ n, let us define a chord diagram Am,n,k such that the chord set of Am,n,k is E(m, n) ∪ {e}, where e intersects no edge of E1 and k edges of E2 . Similarly, for 0 ≤ k ≤ n, let us define a chord diagram Bm,n,k such that the chord set of Bm,n,k is E(m, n)∪{e}, where e intersects all edges of E1 and k edges of E2 . (See Fig.3.) Let am,n,k = f (Am,n,k ) and bm,n,k = f (Bm,n,k ). Note that we have am,n,0 = fm,n , am,n,n = fm+1,n , and bm,n,0 = fm,n+1 .
T. Nakamigawa / Electronic Notes in Discrete Mathematics 54 (2016) 51–56
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Lemma 3.1 For m ≥ 0 and n ≥ k ≥ 1, we have am,n,k = am,n,k−1 + bm,n−1,k−1 , bm,n,k = bm,n,k−1 + am,n−1,k−1 . Proof. Let us label the chords and vertices such that e = zw, E1 = {e1,j = x1,j y1,j : 1 ≤ j ≤ m}, E2 = {e2,j = x2,j y2,j : 1 ≤ j ≤ n}. We may assume the vertices are placed on a circle in clockwise order as follows. For Am,n,k , the order of the vertices is x1,1 , x1,2 , . . . , x1,m , z, x2,1 , x2,2 , . . . , x2,k , w, x2,k+1 , x2,k+2 , . . . , x2,n , y1,m , y1,m−1 , . . . , y1,1 , y2,n , y2,n−1 , . . . , y2,1 , and for Bm,n,k , the order of the vertices is x1,1 , x1,2 , . . . , x1,m , z, x2,1 , x2,2 , . . . , x2,n , y1,m , y1,m−1 , . . . , y1,1 , y2,n , y2,n−1 , . . . , y2,k+1 , w, y2,k , y2,k−1 , . . . , y2,1 . Let us choose a crossing S = {e, e2,k }. For Am,n,k , the expansion with respect to S yields two chord diagrams isomorphic to Am,n,k−1 and Bm,n−1,k−1 , and for Bm,n,k , the expansion with respect to S yields two chord diagrams 2 isomorphic to Bm,n,k−1 and Am,n−1,k−1 . Lemma 3.2 For m ≥ 0 and we have the n ≥ k ≥ 0, following equations. k k (a) am,n,k = am,n−i,0 + bm,n−i,0 . i i i:even,0≤i≤k i:odd,0≤i≤k k k (b) bm,n,k = bm,n−i,0 + am,n−i,0 . i i i:even,0≤i≤k i:odd,0≤i≤k Proof. (a) We proceed by induction on k. For k = 0, bothhandsides of the equation are am,n,0 . Let k ≥ 1. By Lemma 3.1 and the inductive hypothesis, we have am,n,k = am,n,k−1 + bm,n−1,k−1 k−1 k−1 = am,n−i,0 + bm,n−i,0 i i i:even,0≤i≤k−1 i:odd,0≤i≤k−1 k−1 k−1 + bm,n−1−i,0 + am,n−1−i,0 i i i:even,0≤i≤k−1 i:odd,0≤i≤k−1 k − 1 k−1 = am,n−i,0 + am,n−i,0 i i−1 i:even,0≤i≤k−1 i:even,1≤i≤k k − 1 k−1 + bm,n−i,0 + bm,n−i,0 i i − 1 i:odd,0≤i≤k−1 i:odd,1≤i≤k k k = am,n−i,0 + bm,n−i,0 , i i i:even,0≤i≤k i:odd,0≤i≤k
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as desired. By exchanging the roles of am,n,k and bm,n,k , the equation (b) is proved in the same manner. 2 Lastly, we will show the recurrence (2). By Lemma 3.2, we have fm+1,n = am,n,n =
n n am,n−i,0 + bm,n−i,0 i i i:even,0≤i≤n i:odd,0≤i≤n n n = am,n+1−i,0 + bm,n−i,0 i − 1 i i:odd,1≤i≤n+1 i:odd,0≤i≤n n n = ( + )fm,n+1−i i−1 i i:odd,0≤i≤n+1 n+1 = fm,n+1−i . i i:odd,0≤i≤n+1
This completes the proof.
References [1] T. Nakamigawa, Expansions of a chord diagram and alternating permutations, Electronic J. Combinatorics 23 (2016), P1.7. [2] R. P. Stanley, A survey of alternating permutations, Mathematics, 531 (2010), 165–196.
Comtemporary
Electronic Notes in Discrete Mathematics 54 (2016) 51–56
www.elsevier.com/locate/endm
Enumeration Problems on the Expansion of a Chord Diagram Tomoki Nakamigawa
1,2
Department of Information Science Shonan Institute of Technology 1-1-25 Tsujido-Nishikaigan, Fujisawa, Kanagawa 251-8511, Japan
Abstract A chord diagram is a set of chords of a circle such that no pair of chords has a common endvertex. A pair of chords is called a crossing if the two chords intersect. A chord diagram E is called nonintersecting if E contains no crossing. For a chord diagram E having a crossing S = {x1 x3 , x2 x4 }, the expansion of E with respect to S is to replace E with E1 = (E \ S) ∪ {x2 x3 , x4 x1 } or E2 = (E \ S) ∪ {x1 x2 , x3 x4 }. A chord diagram E = E1 ∪ E2 is called complete bipartite of type (m, n), denoted by Cm,n , if (1) both E1 and E2 are nonintersecting, (2) for every pair e1 ∈ E1 and e2 ∈ E2 , e1 and e2 are crossing, and (3) |E1 | = m, |E2 | = n. Let fm,n be the cardinality of the multiset of all nonintersecting chord diagrams generated from Cm,n with a finite sequence of expansions. In this paper, it is shown m,n fm,n (xm /m!)(y n /n!) is 1/(cosh x cosh y − (sinh x + sinh y)). Keywords: enumeration, chord diagram, expansion, alternating permutation
1 2
This work was supported by KAKENHI(16K05260). Email:[email protected]
http://dx.doi.org/10.1016/j.endm.2016.09.010 1571-0653/© 2016 Elsevier B.V. All rights reserved.
52
T. Nakamigawa / Electronic Notes in Discrete Mathematics 54 (2016) 51–56
Fig. 1. The expansion of a chord diagram.
1
Introduction
Let us consider a set of chords of a circle. A set of chords is called a chord diagram, if they have no common endvertex. If a chord diagram consists of a set of n mutually crossing chords, it is called an n-crossing. A 2-crossing is simply called a crossing as well. If a chord diagram contains no crossing, it is called nonintersecting. Let V be a set of 2n vertices on a circle, and let E be a chord diagram of order n, where each chord has endvertices of V . We denote the family of all such chord diagrams by CD(V ). Let x1 , x2 , x3 , x4 ∈ V be placed on a circle in clockwise order. Let E ∈ CD(V ). For a crossing S = {x1 x3 , x2 x4 } ⊂ E, let S1 = {x2 x3 , x4 x1 }, and S2 = {x1 x2 , x3 x4 }. The expansion of E with respect to S is defined as a replacement of E with E1 = (E \ S) ∪ S1 or E2 = (E \ S) ∪ S2 (see Figure 1). For a chord diagram E ∈ CD(V ) and a crossing S ⊂ E, we have two successors E1 and E2 of E. By iterating possible expansions, we have a multiset of nonintersecting chord diagrams. As is remarked in [1], the resulting multiset of nonintersecting chord diagrams generated from E with a set of expansions is uniquely determined. Let us denote the multiset of nonintersecting chord diagrams generated by E ∈ CD(V ) by N CD(E). For E ∈ CD(V ), let us define f (E) as the cardinality of N CD(E) as a multiset. Let Cn be an n-crossing. In [1], the exponential generating function of
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Fig. 2. Complete bipartite chord diagram Cm,n , where m = 4 and n = 3.
f (Cn ) is shown as 1 xn f (Cn ) = n! 1 − sin x n≥0 x3 x4 x5 x6 x2 + 5 + 16 + 61 + 272 + · · · . 2! 3! 4! 5! 6! Hence, f (Cn ) = En+1 , where En is called the Euler number, which corresponds to the number of alternating permutations of order n (see [2] for an excellent survey of alternating permutations). A chord diagram E is called complete bipartite of type (m, n), where m and n are nonnegative integers, if E is paratitioned into E1 and E2 such that (1) both E1 and E2 are nonintersecting, (2) for every pair of chords e1 ∈ E1 and e2 ∈ E2 , e1 and e2 are crossing, and (3) |E1 | = m, |E2 | = n. (See Fig.2.) Let us denote a complete bipartite chord diagram of type (m, n) by Cm,n , and let us denote f (Cm,n ) by fm,n . In this paper, we focus on counting fm,n . =1 + x + 2
2
Main Results
Let define the exponential generating function F (x, y) as us m n m≥0 n≥0 fm,n (x /m!)(y /n!). The main result of the paper is the following theorem. Theorem 2.1 F (x, y) =
1 . cosh x cosh y − (sinh x + sinh y)
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T. Nakamigawa / Electronic Notes in Discrete Mathematics 54 (2016) 51–56
Fig. 3. Am,n,k (left) and Bm,n,k (right), where m = 4, n = 3 and k = 2.
3
Proof of Theorem 2.1
Let us denote the righthandside of the equation of Theorem 2.1 by g(x, y). Then we have g(x, y) = g(y, x), g(x, 0) = ex and ∂ ∂g = (g · sinh y). ∂x ∂y On the other hand, it is easy to see that F (x, y) = F (y, x) and F (x, 0) = ex . It is left for us to check that F (x, y) also satisfies that ∂ ∂F = (F · sinh y). ∂x ∂y Firstly, note that the equation (1) is equivalent to the following recurrence. n+1 (2) fm+1,n = fm,n+1−i i i:odd,1≤i≤n+1 (1)
for all m ≥ 0 and n ≥ 0. In the following, we will show the recurrence (2). Let us introduce new families of chord diagrams. Let E(m, n) = E1 ∪E2 be isomorphic to a complete bipartite chord diagram of type (m, n) with |E1 | = m and |E2 | = n. For 0 ≤ k ≤ n, let us define a chord diagram Am,n,k such that the chord set of Am,n,k is E(m, n) ∪ {e}, where e intersects no edge of E1 and k edges of E2 . Similarly, for 0 ≤ k ≤ n, let us define a chord diagram Bm,n,k such that the chord set of Bm,n,k is E(m, n)∪{e}, where e intersects all edges of E1 and k edges of E2 . (See Fig.3.) Let am,n,k = f (Am,n,k ) and bm,n,k = f (Bm,n,k ). Note that we have am,n,0 = fm,n , am,n,n = fm+1,n , and bm,n,0 = fm,n+1 .
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Lemma 3.1 For m ≥ 0 and n ≥ k ≥ 1, we have am,n,k = am,n,k−1 + bm,n−1,k−1 , bm,n,k = bm,n,k−1 + am,n−1,k−1 . Proof. Let us label the chords and vertices such that e = zw, E1 = {e1,j = x1,j y1,j : 1 ≤ j ≤ m}, E2 = {e2,j = x2,j y2,j : 1 ≤ j ≤ n}. We may assume the vertices are placed on a circle in clockwise order as follows. For Am,n,k , the order of the vertices is x1,1 , x1,2 , . . . , x1,m , z, x2,1 , x2,2 , . . . , x2,k , w, x2,k+1 , x2,k+2 , . . . , x2,n , y1,m , y1,m−1 , . . . , y1,1 , y2,n , y2,n−1 , . . . , y2,1 , and for Bm,n,k , the order of the vertices is x1,1 , x1,2 , . . . , x1,m , z, x2,1 , x2,2 , . . . , x2,n , y1,m , y1,m−1 , . . . , y1,1 , y2,n , y2,n−1 , . . . , y2,k+1 , w, y2,k , y2,k−1 , . . . , y2,1 . Let us choose a crossing S = {e, e2,k }. For Am,n,k , the expansion with respect to S yields two chord diagrams isomorphic to Am,n,k−1 and Bm,n−1,k−1 , and for Bm,n,k , the expansion with respect to S yields two chord diagrams 2 isomorphic to Bm,n,k−1 and Am,n−1,k−1 . Lemma 3.2 For m ≥ 0 and we have the n ≥ k ≥ 0, following equations. k k (a) am,n,k = am,n−i,0 + bm,n−i,0 . i i i:even,0≤i≤k i:odd,0≤i≤k k k (b) bm,n,k = bm,n−i,0 + am,n−i,0 . i i i:even,0≤i≤k i:odd,0≤i≤k Proof. (a) We proceed by induction on k. For k = 0, bothhandsides of the equation are am,n,0 . Let k ≥ 1. By Lemma 3.1 and the inductive hypothesis, we have am,n,k = am,n,k−1 + bm,n−1,k−1 k−1 k−1 = am,n−i,0 + bm,n−i,0 i i i:even,0≤i≤k−1 i:odd,0≤i≤k−1 k−1 k−1 + bm,n−1−i,0 + am,n−1−i,0 i i i:even,0≤i≤k−1 i:odd,0≤i≤k−1 k − 1 k−1 = am,n−i,0 + am,n−i,0 i i−1 i:even,0≤i≤k−1 i:even,1≤i≤k k − 1 k−1 + bm,n−i,0 + bm,n−i,0 i i − 1 i:odd,0≤i≤k−1 i:odd,1≤i≤k k k = am,n−i,0 + bm,n−i,0 , i i i:even,0≤i≤k i:odd,0≤i≤k
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as desired. By exchanging the roles of am,n,k and bm,n,k , the equation (b) is proved in the same manner. 2 Lastly, we will show the recurrence (2). By Lemma 3.2, we have fm+1,n = am,n,n =
n n am,n−i,0 + bm,n−i,0 i i i:even,0≤i≤n i:odd,0≤i≤n n n = am,n+1−i,0 + bm,n−i,0 i − 1 i i:odd,1≤i≤n+1 i:odd,0≤i≤n n n = ( + )fm,n+1−i i−1 i i:odd,0≤i≤n+1 n+1 = fm,n+1−i . i i:odd,0≤i≤n+1
This completes the proof.
References [1] T. Nakamigawa, Expansions of a chord diagram and alternating permutations, Electronic J. Combinatorics 23 (2016), P1.7. [2] R. P. Stanley, A survey of alternating permutations, Mathematics, 531 (2010), 165–196.
Comtemporary