Catalan Traffic and Integrals on the Grassmannians of Lines
aa r X i v : . [ m a t h . AG ] A p r “Catalan Traffic” and Integrals on theGrassmannian of Lines ∗ Ta´ıse Santiago Costa Oliveira † Abstract
We prove that certain numbers occurring in a problem of paths enumeration,studied by Niederhausen in [7] (see also [10]), are top intersection numbers inthe cohomology ring of the grassmannian of the lines in the complex projective( n + 1)-space. The
Catalan’s numbers C n = 1 n + 1 (cid:18) nn (cid:19) , for all n ∈ N occur in several combinatorial situations (see e.g. [9]), in particular in lattice pathenumeration . It is well known, for instance, that C n is the number of lattice pathscontained in S := { ( m, n ) ∈ Z × Z | ≤ m ≤ n } from (0 ,
0) to ( n, n ) ∈ S , allowingunitary steps only, along the “horizontal” or “vertical” directions.Within this context, the aim of this paper is to make some remarks on the oc-currence of Catalan’s numbers in a traffic game (“Catalan traffic at the beach”)constructed by Niederhausen [7]. One is given of a city map C (a lattice in Z ) withsome gates and road blocks (null traffic points). The traffic rules are as follows. First,no path can cross and go beyond the beach (the line m − n = 0 in the ( m, n ) Z -plane).Furthermore: ∗ AMS 2000 Math. Subject Classification: 14M15, 14N15, 05A15, 05A19. † This work was supported in part by ScuDo - Politecnico di Torino, FAPESB proc. n 8057/2006and CNPq proc. n 350259/2006-2.
1. At lattice points strictly “below” the line 2 m + n = 0, only North ( ↑ ) or West( ← ) directions are allowed;2. At lattice points strictly “above” the line 2 m + n = 0, only East ( → ) or NE( ր ) directions are allowed;3. All the points ( m, n ) ∈ Z lying on the line 2 m + n = 1 are inaccessible (roadblocks (cid:4) ).4. On the line 2 m + n = 0 (gates), allow W ( ← ), E ( → ), and N E ( ր ) (becauseof the road blocks at 2 m + n = 1) (see [7], p. 2)The diagram of the city map is depicted below: it is the same as in [7] after aharmless counterclockwise rotation of 90 degrees. − − − − Start here
Fig. 1. City Map.The problem, solved by Niederhausen, consists in finding the number of all dis-tinct paths joining the origin to any point in the domain of C , compatibly with theconstraints. Attaching to each point of the lattice the number of such paths one getsthe following diagram: 2 − − − Fig. 2. Detours preserving Catalan traffic.The main result of [7] is that the numbers along the “beach” are Catalan’s num-ber. This is proven in three different ways. One of them relies on the followingrecursion: (cid:26) Υ( m, n ) = Υ( m + 1 , n ) − Υ( m, n − n, n ) = C n , (1)holding in the domain { ( m, n ) ∈ Z | − n ≤ m ≤ n } , where Υ( m, n ) denotes thenumber of paths to get the point ( m, n ) starting from the origin. The Catalan number C n has also a beautiful geometric interpretation (seee.g. [6]): in fact, it is the Pl¨ucker degree κ n, = Z G ( P n +1 ) σ n , of the grassmannian of lines G ( P n +1 ). The main result of this paper is that for each ( m, n ) , such that n ≥ m ≥
0, Υ( m, n ) = κ m,n − m where κ m,n − m = Z G ( P n +1 ) σ m σ n − m , is the top intersection number computed in the integral cohomology ring H ∗ ( G ( P n +1 ) , Z ) of G ( P n +1 ), generated (as a Z -algebra) by the special Schubertcycles σ and σ . The proof consists in using the formalism introduced in [2] (seealso [3] and [8]) to show that the recursion (1) holds for K ( m, n ) := κ m,n − m . Acknowledgment.
I want to thank my advisor L. Gatto for many helpful dis-cussions and the anonymous referee for ideas and comments.3
Preliminaries
The grassmannian G ( P n +1 ) is a complete projective variety of complex di-mension 2 n . It is known (see e.g. [1], [5]) that its integral cohomology (or Chowintersection ring) H ∗ ( G ( P n ) , Z ) is generated by the special Schubert cycles σ and σ . The cycle σ ∈ H ∗ ( G ( P n +1 ) , Z ) is the cohomology class represented by the sub-variety which is the closure of all the lines incident a codimension 2 linear subspace of P n +1 , while σ ∈ H ∗ ( G ( P n +1 ) , Z ) is represented by the closure of all the lines whichare incident a codimension 3 linear subspace of P n +1 . A top intersection number in G ( P n +1 ) is the degree of the product σ a σ b in H ∗ ( G ( P n +1 ) , Z ), with a + 2 b = 2 n . The cohomology (or the intersection) theory of complex grassmannian varietiescan be described by
Schubert Calculus . The latter can be phrased in purely alge-braic terms using the formalism introduced in [2] (see [3] for more details). For thegrassmannian of lines of P n +1 this works, in short, as follows. Let V M be the 2 nd exterior power of a free module of rank n + 2. If M is spanned by ( ǫ n +1 , ǫ n , . . . , ǫ , ǫ ),then V M is freely generated by { ǫ i ∧ ǫ j | ≤ i < j ≤ n + 1 } .Let D : M −→ M such that D ǫ i = ǫ i − if i > D ε = 0. Extend D toan endomorphism of V M by setting: D ( ǫ i ∧ ǫ j ) = D ǫ i ∧ ǫ j + ǫ i ∧ D ǫ j , and let D : V M −→ V M such that: D ( ǫ i ∧ ǫ j ) = D ǫ i ∧ ǫ j + D ǫ i ∧ D ǫ j + ǫ i ∧ D ǫ j . In other words D behaves as a “first derivative” and D as a “second derivative”.The main result in [2] is that the endomorphism D and D generate a commutativesubalgebra A ∗ of the Z -algebra End Z ( V M ) which is isomorphic to H ∗ ( G ( P n ) , Z ).The isomorphism is explicitly obtained from sending σ D and σ D . Fromthis point of view, it turns out that the degree of a top intersection product σ a σ b ( a + 2 b = 2 n ) in H ∗ ( G ( P n +1 ) , Z ) is nothing else than the coefficient κ a,b in theequality: D a D b ( ǫ n +1 ∧ ǫ n ) = κ a,b · ǫ ∧ ǫ . In this section we will prove the main result of this paper: the connection betweenthe numbers in the Catalan Traffic and top intersection numbers in the integral4ohomology ring of the grassmannian G ( P ) of lines in P . This connection is aconsequence of the following: Let K ( m, n ) := κ m,n − m be the coefficient of ǫ ∧ ǫ in the expansionof D m D n − m ( ε n +1 ∧ ε n ) . Thus, for all ≤ m ≤ n the following recursion hold: K ( m, n ) = K ( m + 1 , n ) − K ( m, n − . Proof.
Let ∆ be the endomorphism of M defined by:∆ ( D )( ǫ i ∧ ǫ j ) = D ǫ i ∧ D ǫ j = ( D − D )( ǫ i ∧ ǫ j ) . (2)Recalling that D m D n − m ( ǫ n +1 ∧ ǫ n ) = K ( m, n ) · ǫ ∧ ǫ , by definition of K ( m, n ), one has K ( m, n ) · ǫ ∧ ǫ = D m D n − m ( ǫ n +1 ∧ ǫ n ) == D m D n − m − ( D − ∆ )( ǫ n +1 ∧ ǫ n ) == D m +21 D n − m − ( ǫ n +1 ∧ ǫ n ) − D m D n − m − ( ǫ n ∧ ǫ n − ) . (3)Now, on the r.h.s of formula (3), the former summand is precisely K ( m + 1 , n ) ǫ ∧ ǫ while the latter, using (2), is equal to D m D n − m − ( ǫ n ∧ ǫ n − ) which in turn equals κ m,n − m − ǫ ∧ ǫ . Hence, keeping in mind that κ m,n − m − = K ( m, n − K ( m, n ) · ǫ ∧ ǫ = ( K ( m + 1 , n ) − K ( m, n − · ǫ ∧ ǫ . As a conclusion K ( m + 1 , n ) = K ( m, n ) + K ( m, n − . For all n ≥ , K ( n, n ) = C n . Proof.
In fact one has: D n ( ǫ n +1 ∧ ǫ n ) = n X i =0 (cid:18) ni (cid:19) D i ǫ n +1 ∧ D n − i ǫ n = n X i =0 (cid:18) ni (cid:19) ǫ n +1 − i ∧ ǫ i − n (4)5ince n ≤ i ≤ n + 1, i = n or i = n + 1. Hence only the sum (cid:18) nn (cid:19) ǫ ∧ ǫ + (cid:18) nn + 1 (cid:19) ǫ ∧ ǫ can survive in expression (4). Therefore: D n ǫ n +1 ∧ ǫ n = (cid:20)(cid:18) nn (cid:19) − (cid:18) nn + 1 (cid:19)(cid:21) ǫ ∧ ǫ (5)so that K ( n, n ) = (cid:18) nn (cid:19) − (cid:18) nn + 1 (cid:19) = (2 n )!( n + 1)! n ! = C n . For all ≤ m ≤ n , the number Υ( m, n ) (see equation (1)) coincideswith the number κ m,n − m of lines in P n +1 incident m linear subspaces of codimension and n − m subspaces of codimension in general position in P n +1 . Proof.
This follows from Theorem 3.1, Proposition 3.2 and the remarks in Section2.
A formula for the number κ a,b , with a + 2 b = 2 n is computed in [8].For a = 2 m and b = n − m , it coincides with the following combinatorial expression: κ m,n − m = n − m X i =0 2 m X j =0 (cid:18) mj (cid:19) ( n − m )!( m + n − j − i + 1) i !( n − j − i + 1)!( i + j − m )! , (6)holding for each n ≥ m ≥ D i , studiedin [8] and not mentioned here, lead to the following simplified form: κ m,n − m = n − m X i =0 ( − i (cid:18) n − mi (cid:19) C n − i , holding for each n ≥ m ≥
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