- Email: [email protected]
Visualizing Vandermonde's determinant through nonintersecting lattice paths
ARTICLE IN PRESS Journal of Statistical Planning and Inference 140 (2010) 2346–2350
Contents lists available at ScienceDirect
Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi
Visualizing Vandermonde’s determinant through nonintersecting lattice paths Jennifer J. Quinn Interdisciplinary Arts and Sciences, University of Washington, Tacoma, 1900 Commerce Street, Box 358436, Tacoma, WA 98402-3100, USA
a r t i c l e i n f o
abstract
Available online 21 January 2010
¨ We use the Lindstrom–Gessel–Viennot Theorem to count nonintersecting lattice paths in a carefully chosen acyclic weighted digraph to give a visual combinatorial proof of Vandermonde’s classic determinant. & 2010 Elsevier B.V. All rights reserved.
Keywords: Combinatorial proofs Signed involution ¨ Lindstrom–Gessel–Viennot
1. Introduction The determinant of the n n Vandermonde matrix, Vn = [xji 1] for 1 r i; j rn; has the beautiful product form 1 1 det Vn ¼ 1 ^ 1
x1
x21
x2
x3
x22 x23
^
^
&
xn
x2n
xn1 1 n1 x2 Y ¼ xn1 ðxj xi Þ: 3 ^ 1 r i o j r n xn1 n
This matrix and its determinant play important roles in polynomial interpolation, partition theory, coding theory, signal processing, and even quantum physics. The original proof (due to Augustin-Louis Cauchy in 1812) exploits symmetry— exchanging two rows and considering the resulting alternating polynomial (Bressoud, 1999). Many proofs have been given since (see, e.g. Bellman, 1987; Blahut, 1983; Pless, 1998) including several clever combinatorial ones (Benjamin and Dresden, 2007; Gessel, 1979). Combinatorial methods often bring deeper understanding to algebraic identities. While there are various combinatorial interpretations of determinants (Benjamin and Cameron, 2005; Brualdi, 1991; Zeilberger, 1985), we choose to use the ¨ ¨ Lindstrom–Gessel–Viennot Theorem (Gessel and Viennot, 1985; Lindstrom, 1973) to count nonintersecting lattice paths. This result has been independently rediscovered in several different mathematical settings and we recommend Krattenthaler’s (2006) excellent account for a complete history. With the restriction that the xi’s are strictly increasing, the Vandermonde matrix is one example of a totally positive matrix, that is to say, a matrix all of whose minors are positive. Brenti (1995) proved that the determinant of a totally positive matrix always corresponds to a (nontrivial) weighted digraph whose nonintersecting lattice paths realize the determinant. This paper constructs such a digraph for the Vandermonde matrix, thus completing the story.
E-mail address: [email protected] 0378-3758/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2010.01.029
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2. Preliminaries Let D= (V,E) be a directed acyclic graph with vertex set V and directed edges E. Suppose we are given a function w that Q assigns a weight w(e) to every edge e 2 E. If P is a path from vertex vi to vj, define the weight of P by e2P wðeÞ, the product of the weights of the edges in P. The total weight from vi to vj, denoted tw(vi, vj), is then defined as the sum of the weights over all paths from vi to vj. Since the digraph is acyclic, the total weight between two vertices is well-defined. When i =j, we have the total weight of the empty paths is tw(vi, vi)=1. ¨ (1973) reveal how the determinant of a matrix counts signed nonintersecting Gessel and Viennot (1985) and Lindstrom path-systems in an associated directed graph. For an n n matrix A = [aij], the general idea is to create an acyclic directed graph D with n origin nodes, o1, o2, y, on, n destination nodes, d1, d2, y, dn, and a weight function w, so that the total Q weight from origin oi to destination dj is aij. Given a permutation s in Sn, the product ni¼ 1 aisðiÞ represents the total weight of n directed paths in D where the i th path goes from origin oi to destination dsðiÞ . We call such a system of n directed paths Q P a weighted n-route. Since detðAÞ ¼ s2Sn sgnðsÞ ni¼ 1 aisðiÞ , where sgnðsÞ is the sign of the permutation, the determinant is the sum of weighted n-routes induced by even permutations (called even n-routes) minus the sum of weighted n-routes induced by odd permutations (called odd n-routes). A sign reversing involution exists between even and odd n-routes provided some vertex of D is shared by two paths in the route. That is to say, given any self-intersecting n-route, there is an n-route of opposite parity that uses exactly the same edges. So calculating the determinant reduces to determining the weight of even nonintersecting n-routes minus the weight of odd nonintersecting n-routes. For a quick illustration, the matrix 2 3 2 3 8 6 7 A ¼ 4 4 7 18 5 ð1Þ 5 7 19 corresponds to the weighted digraph given in Fig. 1. Clearly, the total weights from origins o1, o2, and o3 to destination d1 are 2, 4, and 5, respectively. There are three different paths from o3 to d2 of weights 3, 2, and 2. So tw(o3, d2)=7 as desired. The reader is encouraged to verify that the remaining entries correspond to total weights in the digraph. Since the three origins have only two distinct neighbors, any 3-route will necessarily intersect at one of those two points. Having no nonintersecting ¨ n-routes (either even or odd), the Linstrom–Gessel–Viennot Theorem requires that the determinant of A equals zero. The matrix in (1) is an example of the more general identity 2 3 Gm Gm þ p Gm þ q 6 7 det4 Hm þ r Hm þ p þ r Hm þ q þ r 5 ¼ 0 Im þ s Im þ p þ s Im þ q þ s with G, H, and I being generalized Fibonacci sequences—the next element in the sequence is found by summing the previous two beginning with arbitrary initial conditions G0, G1, H0, H1, I0, and I1 (Benjamin et al., 2007). It must be noted that any determinant can be realized by some weighted directed graph. In fact, we can always use a complete bipartite graph and weight the single edge from origin oi to destination dj by the ij th matrix entry. However, computing the determinant by nonintersecting n-routes in this case would be equivalent to expanding the determinant over all permutations since all n-routes are nonintersecting. This might not bring any deeper insights. The goal is to find a digraph representation that clearly visualizes the determinant. 3. Weighting the integer lattice Given an integer n Z 1; construct the acyclic digraph Dn as follows: (1) the vertices of Dn are the integer points (i,j) where 0 ri; j rn1; (2) the arcs of Dn create the grid on the integer lattice and are directed up or to the right;
Fig. 1. All edges in the digraph are weighted by 1 unless otherwise noted. Origin nodes are enclosed by circles and destination nodes are enclosed by squares. There are no nonintersecting 3-routes, so the corresponding determinant must be zero.
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Fig. 2. D4 and its associated matrix A4.
Fig. 3. The only nonintersecting 4-route for the digraph given in Fig. 2 is associated with the identity permutation. Thus the determinant equals 1.
(3) origin oi is the vertex (0, n i) for 1 r i rn; (4) destination dj is the vertex (j 1, n 1) for 1 rj r n. See Fig. 2. If we assign a weight of 1 to all the edges, the total weight from oi to dj equals the number of paths between j2 them. It takes j 1 horizontal steps and i 1 vertical steps to go from oi to dj, so twðoi ; dj Þ ¼ i þi1 for 1 r rn. i; jj2 . There is Thus Dn and the weight function w(e)= 1 for all e 2 E can be used to compute the determinant of An ¼ ½ i þi1 only one nonintersecting n-route and it corresponds to the identity permutation (see Fig. 3). Since the identity permutation is even, det(An) = 1. Changing the weight function on Dn changes the associated matrix. With the Vandermonde determinant as our target, the motivation for assigning weights becomes evident. There are as many horizontal edges below the main diagonal as Q there are factors in 1 r i o j r n ðxj xi Þ, so assigning one factor as the weight of each such horizontal edge and a weight of 1 to each vertical edge will yield the desired product as the determinant of the corresponding matrix. Furthermore, we will have complete freedom to weight the horizontal edges above the main diagonal to force the associated matrix to be the Vandermonde matrix. Specifically, we use the weighting described in Table 1 and illustrated in Fig. 4. Clearly the Q determinant of the corresponding matrix will be 1 r i o j r n ðxj xi Þ. It remains to show that the matrix actually corresponds to the Vandermonde matrix. 4. Vanquishing Vandermonde To show that the total weight between origin oi and destination dj is xji 1, we will, in fact, determine the total weight between an origin and any vertex vs,t =(t 1, n s). For convenience, the index on the vertices will indicate the origin os and the destination dt in the same row and column.
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Table 1 For a positive integer n, the directed edges of Dn are assigned the weight function w depending on the description of the edge as vertical, horizontal and below the main diagonal, or horizontal and above the main diagonal. For convenience, let vs,t denote vertex (t 1, n s). Description
Form
Weight
Vertical edge
e =((i 1, n j 1), (i 1, n j)) =(vj + 1,i, vj,i) for 1 r i; j r n e = ((i 1, n j), (i, n j)) =(vj,i, vj,i + 1) for 1 r io j r n e =((i 1, n j), (i, n j)) = (vj,i, vj,i + 1) for 1r jr ir n
w(e) = 1
Horizontal below main diagonal Horizontal above main diagonal
w(e) = xj xj i w(e) = xj
Fig. 4. With horizontal weights as indicated and vertical weights equal to 1, the determinant of the matrix corresponding to the signed sum of the Q nonintersecting weighted n-routes of this directed graph equals 1 r i o j r n ðxj xi Þ.
Theorem 1. For 1 r sr i rn, 1r t rn, the total weight from origin oi to vertex vs,t = (t 1, n s) in digraph Dn with weights as given in Table 1 is 8Q < t1 ðxi xs‘ Þ for s Zt; ‘¼1 Qs1 : xts ðx x Þ for s ot; s‘ ‘¼1 i i where we define
Q0
‘ ¼ 1 ðxi xs‘ Þ ¼
1.
Proof. The proof will be by nested induction. The statement of the theorem clearly holds for small fixed values of n. Assume that the statement holds for a weighted n n lattice for some n Z2: To show that the statement holds for an ðn þ 1Þ ðn þ 1Þ lattice, we need only verify the total weights from origins oi ð1 r ir nÞ to vertices of the form vs, n + 1 and the total weights from on + 1 to arbitrary vertices of the graph. Case 1: Total weight from oi to vs,n + 1 ð1 r s ri rnÞ. Proceed by induction on i s. If s= i (so i s= 0), the graph was Q Q n þ 1s s1 constructed so that twðoi ; vi;n þ 1 Þ ¼ xni þ 1i i1 ‘ ¼ 1 ðxi xi‘ Þ. Assume that twðoi ; vs;n þ 1 Þ ¼ xi ‘ ¼ 1 ðxi xs‘ Þ whenever 0 r is o k, for some k and suppose is ¼ k. The total weight from oi to vs,n + 1 is the sum of the total weight from oi to vs + 1,n + 1 plus xs times the total weight from oi to vs,n. Each of these component weights are known by induction, so twðoi ; vs;n þ 1 Þ ¼ twðoi ; vs þ 1;n þ 1 Þ þ xs twðoi ; vs;n Þ ¼ xns i
s Y
ðxi xs þ 1‘ Þ þ xs xns i
‘¼1
¼ ½ðxi xs Þ þ ðxs Þ xns i
s1 Y ‘¼1
as desired.
ðxi xs‘ Þ ¼ xni þ 1s
s1 Y ‘¼1
s1 Y
ðxi xs‘ Þ
‘¼1
ðxi xs‘ Þ
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Case 2: Total weight from on + 1 to vs,t. By construction, tw(on + 1, vs,1)= 1 for all 1 r s rn þ1 and twðon þ 1 ; vn þ 1;t Þ ¼ Qt1 ‘ ¼ 1 ðxn þ 1 xn þ 1‘ Þ for all 1 rt r n þ 1. To verify the total weight from on + 1 to an arbitrary vertex vb,c, assume that 8 t1 Y > > > ðxn þ 1 xs‘ Þ for s Zt; > > <‘¼1 twðon þ 1 ; vs;t Þ ¼ ð2Þ s1 Y > > > > xts ðxn þ 1 xs‘ Þ for s ot > n þ 1 : ‘¼1
holds for all values of s provided 1 rt o c and for b o s rn þ1 provided t= c. As we saw before, twðon þ 1 ; vb;c Þ ¼ twðon þ 1 ; vb þ 1;c Þ þwðvb;c1 ; vb;c Þ twðon þ 1 ; vb;c1 Þ: The weight of the edge (vb,c 1, vb,c) depends on whether or not b is greater than or equal to c. If b Zc, then w(vb,c 1, vb,c)= xb xb c + 1. Invoking induction to compute the total weight gives twðon þ 1 ; vb;c Þ ¼
cY 1
ðxn þ 1 xb þ 1‘ Þ þ ðxb xbc þ 1 Þ
‘¼1
cY 2
ðxn þ 1 xb‘ Þ
‘¼1
¼ ðxn þ 1 xb Þðxn þ 1 xb1 Þ ðxn þ 1 xbc þ 2 Þ þ ðxb xbc þ 1 Þðxn þ 1 xb1 Þ ðxn þ 1 xbc þ 2 Þ cY 2 cY 1 ðxn þ 1 xb‘ Þ ¼ ðxn þ 1 xb‘ Þ ¼ ½ðxn þ 1 xb Þ þ ðxb xbc þ 1 Þ ‘¼1
‘¼1
as desired. If b o c, then w(vb,c 1, vb,c)= xb. The verification when bo c1 mirrors the computation given in case 1. However, care must be taken to verify the special instance when b= c 1 separately: twðon þ 1 ; vb;b þ 1 Þ ¼
b Y
ðxn þ 1 xb þ 1‘ Þ þxb
‘¼1
¼ ½ðxn þ 1 xb Þ þ ðxb Þ
b1 Y
ðxn þ 1 xb‘ Þ ¼ ðxn þ 1 xb Þðxn þ 1 xb1 Þ ðxn þ 1 x1 Þ þ ðxb Þðxn þ 1 xb1 Þ ðxn þ 1 x1 Þ
‘¼1 b1 Y ‘¼1
ðxn þ 1 xb‘ Þ ¼ xn þ 1
b1 Y
ðxn þ 1 xb‘ Þ:
‘¼1
Since the total weights between an origin oi and an arbitrary vertex vs,t in the ðn þ1Þ ðn þ 1Þ weighted lattice have the desired values, the theorem holds for all positive integers n by induction. & The immediate consequence of Theorem 1 is that tw(oi, dj) = xj 1, 1 ri; j r n. So the integer lattice weighted with the function given in Table 1 does indeed represent the Vandermonde determinant and we can easily ‘‘see’’ that the weight of Q the only nonintersecting route is 1 r i o j r n ðxj xi Þ. In the future, we hope to find a more direct (ideally combinatorially) verification that the weighted integer lattice corresponds to Vandermonde’s determinant and we would like to investigate other weightings of the integer lattice—perhaps with an eye to finding a visualization of Krattenthaler’s generalization of Vandermonde’s determinant. References R. Bellman, Introduction to Matrix Analysis. Society for Industrial and Applied Mathematics, 1987. Benjamin, A.T., Cameron, N.T., 2005. Counting on determinants. Amer. Math. Monthly 112, 481–492. Benjamin, A.T., Cameron, N.T., Quinn, J.J., 2007. Fibonacci determinants—a combinatorial approach. Fibonacci Quarterly 45, 39–55. Benjamin, A.T., Dresden, G.P., 2007. A combinatorial proof of Vandermonde’s determinant. Amer. Math. Monthly 114, 338–341. Blahut, R.E., 1983. Theory and Practice of Error Control Codes. Addison-Wesley, Reading, MA. Brenti, F., 1995. Combinatorics and total positivity. J. Combin. Theory Ser. A 71 (2), 175–218. Bressoud, D.M., 1999. Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture. Mathematical Association of America, Washington, DC. Brualdi, R.A., 1991. Combinatorial Matrix Theory. Cambridge University Press, New York. Gessel, I., 1979. Tournaments and Vandermonde’s determinant. J. Graph Theory 3, 305–307. Gessel, I., Viennot, X.G., 1985. Binomial determinants, paths, and hook length formulae. Adv. Math. 58 (3), 300–321. Krattenthaler, C., 2006. Watermelon configurations with wall interaction: exact and asymptotic results. J. Phys. Conf. Ser. 42, 179–212. ¨ Lindstrom, B., 1973. On the vector representations of induced matroids. Bull. London Math. Soc. 5, 85–90. Pless, V., 1998. Introduction to the Theory of Error-Correcting Codes. Wiley-Interscience, New York. Zeilberger, D., 1985. A combinatorial approach to matrix algebra. Discrete Math. 56, 61–72.
Contents lists available at ScienceDirect
Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi
Visualizing Vandermonde’s determinant through nonintersecting lattice paths Jennifer J. Quinn Interdisciplinary Arts and Sciences, University of Washington, Tacoma, 1900 Commerce Street, Box 358436, Tacoma, WA 98402-3100, USA
a r t i c l e i n f o
abstract
Available online 21 January 2010
¨ We use the Lindstrom–Gessel–Viennot Theorem to count nonintersecting lattice paths in a carefully chosen acyclic weighted digraph to give a visual combinatorial proof of Vandermonde’s classic determinant. & 2010 Elsevier B.V. All rights reserved.
Keywords: Combinatorial proofs Signed involution ¨ Lindstrom–Gessel–Viennot
1. Introduction The determinant of the n n Vandermonde matrix, Vn = [xji 1] for 1 r i; j rn; has the beautiful product form 1 1 det Vn ¼ 1 ^ 1
x1
x21
x2
x3
x22 x23
^
^
&
xn
x2n
xn1 1 n1 x2 Y ¼ xn1 ðxj xi Þ: 3 ^ 1 r i o j r n xn1 n
This matrix and its determinant play important roles in polynomial interpolation, partition theory, coding theory, signal processing, and even quantum physics. The original proof (due to Augustin-Louis Cauchy in 1812) exploits symmetry— exchanging two rows and considering the resulting alternating polynomial (Bressoud, 1999). Many proofs have been given since (see, e.g. Bellman, 1987; Blahut, 1983; Pless, 1998) including several clever combinatorial ones (Benjamin and Dresden, 2007; Gessel, 1979). Combinatorial methods often bring deeper understanding to algebraic identities. While there are various combinatorial interpretations of determinants (Benjamin and Cameron, 2005; Brualdi, 1991; Zeilberger, 1985), we choose to use the ¨ ¨ Lindstrom–Gessel–Viennot Theorem (Gessel and Viennot, 1985; Lindstrom, 1973) to count nonintersecting lattice paths. This result has been independently rediscovered in several different mathematical settings and we recommend Krattenthaler’s (2006) excellent account for a complete history. With the restriction that the xi’s are strictly increasing, the Vandermonde matrix is one example of a totally positive matrix, that is to say, a matrix all of whose minors are positive. Brenti (1995) proved that the determinant of a totally positive matrix always corresponds to a (nontrivial) weighted digraph whose nonintersecting lattice paths realize the determinant. This paper constructs such a digraph for the Vandermonde matrix, thus completing the story.
E-mail address: [email protected] 0378-3758/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2010.01.029
ARTICLE IN PRESS J.J. Quinn / Journal of Statistical Planning and Inference 140 (2010) 2346–2350
2347
2. Preliminaries Let D= (V,E) be a directed acyclic graph with vertex set V and directed edges E. Suppose we are given a function w that Q assigns a weight w(e) to every edge e 2 E. If P is a path from vertex vi to vj, define the weight of P by e2P wðeÞ, the product of the weights of the edges in P. The total weight from vi to vj, denoted tw(vi, vj), is then defined as the sum of the weights over all paths from vi to vj. Since the digraph is acyclic, the total weight between two vertices is well-defined. When i =j, we have the total weight of the empty paths is tw(vi, vi)=1. ¨ (1973) reveal how the determinant of a matrix counts signed nonintersecting Gessel and Viennot (1985) and Lindstrom path-systems in an associated directed graph. For an n n matrix A = [aij], the general idea is to create an acyclic directed graph D with n origin nodes, o1, o2, y, on, n destination nodes, d1, d2, y, dn, and a weight function w, so that the total Q weight from origin oi to destination dj is aij. Given a permutation s in Sn, the product ni¼ 1 aisðiÞ represents the total weight of n directed paths in D where the i th path goes from origin oi to destination dsðiÞ . We call such a system of n directed paths Q P a weighted n-route. Since detðAÞ ¼ s2Sn sgnðsÞ ni¼ 1 aisðiÞ , where sgnðsÞ is the sign of the permutation, the determinant is the sum of weighted n-routes induced by even permutations (called even n-routes) minus the sum of weighted n-routes induced by odd permutations (called odd n-routes). A sign reversing involution exists between even and odd n-routes provided some vertex of D is shared by two paths in the route. That is to say, given any self-intersecting n-route, there is an n-route of opposite parity that uses exactly the same edges. So calculating the determinant reduces to determining the weight of even nonintersecting n-routes minus the weight of odd nonintersecting n-routes. For a quick illustration, the matrix 2 3 2 3 8 6 7 A ¼ 4 4 7 18 5 ð1Þ 5 7 19 corresponds to the weighted digraph given in Fig. 1. Clearly, the total weights from origins o1, o2, and o3 to destination d1 are 2, 4, and 5, respectively. There are three different paths from o3 to d2 of weights 3, 2, and 2. So tw(o3, d2)=7 as desired. The reader is encouraged to verify that the remaining entries correspond to total weights in the digraph. Since the three origins have only two distinct neighbors, any 3-route will necessarily intersect at one of those two points. Having no nonintersecting ¨ n-routes (either even or odd), the Linstrom–Gessel–Viennot Theorem requires that the determinant of A equals zero. The matrix in (1) is an example of the more general identity 2 3 Gm Gm þ p Gm þ q 6 7 det4 Hm þ r Hm þ p þ r Hm þ q þ r 5 ¼ 0 Im þ s Im þ p þ s Im þ q þ s with G, H, and I being generalized Fibonacci sequences—the next element in the sequence is found by summing the previous two beginning with arbitrary initial conditions G0, G1, H0, H1, I0, and I1 (Benjamin et al., 2007). It must be noted that any determinant can be realized by some weighted directed graph. In fact, we can always use a complete bipartite graph and weight the single edge from origin oi to destination dj by the ij th matrix entry. However, computing the determinant by nonintersecting n-routes in this case would be equivalent to expanding the determinant over all permutations since all n-routes are nonintersecting. This might not bring any deeper insights. The goal is to find a digraph representation that clearly visualizes the determinant. 3. Weighting the integer lattice Given an integer n Z 1; construct the acyclic digraph Dn as follows: (1) the vertices of Dn are the integer points (i,j) where 0 ri; j rn1; (2) the arcs of Dn create the grid on the integer lattice and are directed up or to the right;
Fig. 1. All edges in the digraph are weighted by 1 unless otherwise noted. Origin nodes are enclosed by circles and destination nodes are enclosed by squares. There are no nonintersecting 3-routes, so the corresponding determinant must be zero.
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Fig. 2. D4 and its associated matrix A4.
Fig. 3. The only nonintersecting 4-route for the digraph given in Fig. 2 is associated with the identity permutation. Thus the determinant equals 1.
(3) origin oi is the vertex (0, n i) for 1 r i rn; (4) destination dj is the vertex (j 1, n 1) for 1 rj r n. See Fig. 2. If we assign a weight of 1 to all the edges, the total weight from oi to dj equals the number of paths between j2 them. It takes j 1 horizontal steps and i 1 vertical steps to go from oi to dj, so twðoi ; dj Þ ¼ i þi1 for 1 r rn. i; jj2 . There is Thus Dn and the weight function w(e)= 1 for all e 2 E can be used to compute the determinant of An ¼ ½ i þi1 only one nonintersecting n-route and it corresponds to the identity permutation (see Fig. 3). Since the identity permutation is even, det(An) = 1. Changing the weight function on Dn changes the associated matrix. With the Vandermonde determinant as our target, the motivation for assigning weights becomes evident. There are as many horizontal edges below the main diagonal as Q there are factors in 1 r i o j r n ðxj xi Þ, so assigning one factor as the weight of each such horizontal edge and a weight of 1 to each vertical edge will yield the desired product as the determinant of the corresponding matrix. Furthermore, we will have complete freedom to weight the horizontal edges above the main diagonal to force the associated matrix to be the Vandermonde matrix. Specifically, we use the weighting described in Table 1 and illustrated in Fig. 4. Clearly the Q determinant of the corresponding matrix will be 1 r i o j r n ðxj xi Þ. It remains to show that the matrix actually corresponds to the Vandermonde matrix. 4. Vanquishing Vandermonde To show that the total weight between origin oi and destination dj is xji 1, we will, in fact, determine the total weight between an origin and any vertex vs,t =(t 1, n s). For convenience, the index on the vertices will indicate the origin os and the destination dt in the same row and column.
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2349
Table 1 For a positive integer n, the directed edges of Dn are assigned the weight function w depending on the description of the edge as vertical, horizontal and below the main diagonal, or horizontal and above the main diagonal. For convenience, let vs,t denote vertex (t 1, n s). Description
Form
Weight
Vertical edge
e =((i 1, n j 1), (i 1, n j)) =(vj + 1,i, vj,i) for 1 r i; j r n e = ((i 1, n j), (i, n j)) =(vj,i, vj,i + 1) for 1 r io j r n e =((i 1, n j), (i, n j)) = (vj,i, vj,i + 1) for 1r jr ir n
w(e) = 1
Horizontal below main diagonal Horizontal above main diagonal
w(e) = xj xj i w(e) = xj
Fig. 4. With horizontal weights as indicated and vertical weights equal to 1, the determinant of the matrix corresponding to the signed sum of the Q nonintersecting weighted n-routes of this directed graph equals 1 r i o j r n ðxj xi Þ.
Theorem 1. For 1 r sr i rn, 1r t rn, the total weight from origin oi to vertex vs,t = (t 1, n s) in digraph Dn with weights as given in Table 1 is 8Q < t1 ðxi xs‘ Þ for s Zt; ‘¼1 Qs1 : xts ðx x Þ for s ot; s‘ ‘¼1 i i where we define
Q0
‘ ¼ 1 ðxi xs‘ Þ ¼
1.
Proof. The proof will be by nested induction. The statement of the theorem clearly holds for small fixed values of n. Assume that the statement holds for a weighted n n lattice for some n Z2: To show that the statement holds for an ðn þ 1Þ ðn þ 1Þ lattice, we need only verify the total weights from origins oi ð1 r ir nÞ to vertices of the form vs, n + 1 and the total weights from on + 1 to arbitrary vertices of the graph. Case 1: Total weight from oi to vs,n + 1 ð1 r s ri rnÞ. Proceed by induction on i s. If s= i (so i s= 0), the graph was Q Q n þ 1s s1 constructed so that twðoi ; vi;n þ 1 Þ ¼ xni þ 1i i1 ‘ ¼ 1 ðxi xi‘ Þ. Assume that twðoi ; vs;n þ 1 Þ ¼ xi ‘ ¼ 1 ðxi xs‘ Þ whenever 0 r is o k, for some k and suppose is ¼ k. The total weight from oi to vs,n + 1 is the sum of the total weight from oi to vs + 1,n + 1 plus xs times the total weight from oi to vs,n. Each of these component weights are known by induction, so twðoi ; vs;n þ 1 Þ ¼ twðoi ; vs þ 1;n þ 1 Þ þ xs twðoi ; vs;n Þ ¼ xns i
s Y
ðxi xs þ 1‘ Þ þ xs xns i
‘¼1
¼ ½ðxi xs Þ þ ðxs Þ xns i
s1 Y ‘¼1
as desired.
ðxi xs‘ Þ ¼ xni þ 1s
s1 Y ‘¼1
s1 Y
ðxi xs‘ Þ
‘¼1
ðxi xs‘ Þ
ARTICLE IN PRESS 2350
J.J. Quinn / Journal of Statistical Planning and Inference 140 (2010) 2346–2350
Case 2: Total weight from on + 1 to vs,t. By construction, tw(on + 1, vs,1)= 1 for all 1 r s rn þ1 and twðon þ 1 ; vn þ 1;t Þ ¼ Qt1 ‘ ¼ 1 ðxn þ 1 xn þ 1‘ Þ for all 1 rt r n þ 1. To verify the total weight from on + 1 to an arbitrary vertex vb,c, assume that 8 t1 Y > > > ðxn þ 1 xs‘ Þ for s Zt; > > <‘¼1 twðon þ 1 ; vs;t Þ ¼ ð2Þ s1 Y > > > > xts ðxn þ 1 xs‘ Þ for s ot > n þ 1 : ‘¼1
holds for all values of s provided 1 rt o c and for b o s rn þ1 provided t= c. As we saw before, twðon þ 1 ; vb;c Þ ¼ twðon þ 1 ; vb þ 1;c Þ þwðvb;c1 ; vb;c Þ twðon þ 1 ; vb;c1 Þ: The weight of the edge (vb,c 1, vb,c) depends on whether or not b is greater than or equal to c. If b Zc, then w(vb,c 1, vb,c)= xb xb c + 1. Invoking induction to compute the total weight gives twðon þ 1 ; vb;c Þ ¼
cY 1
ðxn þ 1 xb þ 1‘ Þ þ ðxb xbc þ 1 Þ
‘¼1
cY 2
ðxn þ 1 xb‘ Þ
‘¼1
¼ ðxn þ 1 xb Þðxn þ 1 xb1 Þ ðxn þ 1 xbc þ 2 Þ þ ðxb xbc þ 1 Þðxn þ 1 xb1 Þ ðxn þ 1 xbc þ 2 Þ cY 2 cY 1 ðxn þ 1 xb‘ Þ ¼ ðxn þ 1 xb‘ Þ ¼ ½ðxn þ 1 xb Þ þ ðxb xbc þ 1 Þ ‘¼1
‘¼1
as desired. If b o c, then w(vb,c 1, vb,c)= xb. The verification when bo c1 mirrors the computation given in case 1. However, care must be taken to verify the special instance when b= c 1 separately: twðon þ 1 ; vb;b þ 1 Þ ¼
b Y
ðxn þ 1 xb þ 1‘ Þ þxb
‘¼1
¼ ½ðxn þ 1 xb Þ þ ðxb Þ
b1 Y
ðxn þ 1 xb‘ Þ ¼ ðxn þ 1 xb Þðxn þ 1 xb1 Þ ðxn þ 1 x1 Þ þ ðxb Þðxn þ 1 xb1 Þ ðxn þ 1 x1 Þ
‘¼1 b1 Y ‘¼1
ðxn þ 1 xb‘ Þ ¼ xn þ 1
b1 Y
ðxn þ 1 xb‘ Þ:
‘¼1
Since the total weights between an origin oi and an arbitrary vertex vs,t in the ðn þ1Þ ðn þ 1Þ weighted lattice have the desired values, the theorem holds for all positive integers n by induction. & The immediate consequence of Theorem 1 is that tw(oi, dj) = xj 1, 1 ri; j r n. So the integer lattice weighted with the function given in Table 1 does indeed represent the Vandermonde determinant and we can easily ‘‘see’’ that the weight of Q the only nonintersecting route is 1 r i o j r n ðxj xi Þ. In the future, we hope to find a more direct (ideally combinatorially) verification that the weighted integer lattice corresponds to Vandermonde’s determinant and we would like to investigate other weightings of the integer lattice—perhaps with an eye to finding a visualization of Krattenthaler’s generalization of Vandermonde’s determinant. References R. Bellman, Introduction to Matrix Analysis. Society for Industrial and Applied Mathematics, 1987. Benjamin, A.T., Cameron, N.T., 2005. Counting on determinants. Amer. Math. Monthly 112, 481–492. Benjamin, A.T., Cameron, N.T., Quinn, J.J., 2007. Fibonacci determinants—a combinatorial approach. Fibonacci Quarterly 45, 39–55. Benjamin, A.T., Dresden, G.P., 2007. A combinatorial proof of Vandermonde’s determinant. Amer. Math. Monthly 114, 338–341. Blahut, R.E., 1983. Theory and Practice of Error Control Codes. Addison-Wesley, Reading, MA. Brenti, F., 1995. Combinatorics and total positivity. J. Combin. Theory Ser. A 71 (2), 175–218. Bressoud, D.M., 1999. Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture. Mathematical Association of America, Washington, DC. Brualdi, R.A., 1991. Combinatorial Matrix Theory. Cambridge University Press, New York. Gessel, I., 1979. Tournaments and Vandermonde’s determinant. J. Graph Theory 3, 305–307. Gessel, I., Viennot, X.G., 1985. Binomial determinants, paths, and hook length formulae. Adv. Math. 58 (3), 300–321. Krattenthaler, C., 2006. Watermelon configurations with wall interaction: exact and asymptotic results. J. Phys. Conf. Ser. 42, 179–212. ¨ Lindstrom, B., 1973. On the vector representations of induced matroids. Bull. London Math. Soc. 5, 85–90. Pless, V., 1998. Introduction to the Theory of Error-Correcting Codes. Wiley-Interscience, New York. Zeilberger, D., 1985. A combinatorial approach to matrix algebra. Discrete Math. 56, 61–72.