Counting Lattice Chains and Delannoy Numbers

Counting Lattice Chains and Delannoy Numbers
A triumph of paper & pencil John Caughman Portland State University Reed College - March 11, 2010

Collaborators 2-dimensional case: Peter Veerman (PSU),
Cliff Haithcock (PSU grad student) Arbitrary dimensional case: Nancy Neudauer (Pacific University) Chuck Dunn (Linfield College), Colin Starr (Willamette University),

What Is a Delannoy number?

Henri Auguste Delannoy
Bourbonne-les-Bains, France. Acquaintance of Edouard Lucas Attended Ecole Polytechnique Several works survive concerning chessboards, Ballot problems, constrained walks (directed Walks of queens), trees in chemistry, some probability, many problems/solutions, watercolor paintings, Many archeological and historical works: “On the significance of the word ieuru” “A bigamist in Gueret” “An impotence trial in the 18th century” Researchers in temporal representation and reasoning use models based on formal languages and use them to enumerate all possible temporal relations between independent event-chronologies.

Delannoy paths in 2-D B What are we counting!? Well TWO things… Delannoy paths, and lattice chains. Let’s look at Delannoy paths in 2D. Begin w/ Rectangular grid…. This is our Integer lattice. In this case, 6 by 4. Want to walk from point A to point B. A

Delannoy paths in L(6,4) y x (6,4) (0,0)
Embed the grid into xy plane, try to get from 00 to 64. We keep track of which grid points we Visit along the way. What makes such a path a Delannoy path is the kinds of steps that are allowed. (0,0) x

Delannoy paths in L(6,4) For a Delannoy path in 2D, we are allowed to move up, right, or diagonally… 3 choices.

Delannoy paths in L(6,4)

Delannoy paths in L(6,4) (6,4) (0,0)

Delannoy paths in L(6,4) (6,4) (0,0) Length 9 1 1 2 1 3 2 3 3 4 5 4 6
1 1 2 1 3 2 3 3 4 5 4 6 4

1 1 2 1 3 2 3 3 4 5 4 6 4 8 steps

Delannoy paths in L(6,4) (6,4) (0,0) Length 8 1 2 1 2 3 4 3 5 4 6 4
1 2 1 2 3 4 3 5 4 6 4 7 steps

Delannoy paths in L(6,4) (6,4)
Delannoy paths are a special case of a lattice chain The x-coordinates form a non-decreasing sequence between 0 and 6 The y-coordinates form a non-decreasing sequence between 0 and 4 (0,0) Length 9 1 1 2 1 3 1 4 2 4 3 5 4 6 4 8 steps

How many Delannoy paths?
We break the counting into 3 cases, depending on the last step… Let D(6,4) = # of Del. paths in L(6,4)

# of Delannoy paths in L(6,4)

# of Delannoy paths in L(6,4)
D(5,4) + D(6,3) + D(5,3) RECURRENCE D(5,3)

Recursively Computed D(m,n)
1 2 3 4 5 6 7 9 11 13 25 41 61 85 63 129 231 377 321 681 1289 1683 3653 8989

An Explicit Formula? Recursively Computed D(m,n) 1 2 3 4 5 6 7 9 11 13
1 2 3 4 5 6 7 9 11 13 25 41 61 85 63 129 231 377 321 681 1289 1683 3653 8989 An Explicit Formula?

What Is a Lattice chain?

Henri Auguste Lattice-Chain
Bourbonne-les-Bains, France. Acquaintance of Edouard Lucas Attended Ecole Polytechnique Several works survive concerning chessboards, Ballot problems, constrained walks (directed Walks of queens), trees in chemistry, some probability, many problems/solutions, watercolor paintings, Many archeological and historical works: “On the significance of the word ieuru” “A bigamist in Gueret” “An impotence trial in the 18th century” Researchers in temporal representation and reasoning use models based on formal languages and use them to enumerate all possible temporal relations between independent event-chronologies.

Partial orders vs. Total orders
Reflexive: For all a, a ≤ a. Antisymmetric: For distinct a,b, a ≤ b implies b ≤ a Transitive: For all a,b,c, a ≤ b and b ≤ c implies a ≤ c Comparability: For all a,b, a ≤ b or b ≤ a Partial order Total order

In a partial order, a chain is a subset that is totally ordered. Partial order

g In a partial order, a chain is a subset that is totally ordered. f {a, b, c, d, e, f, g} is a chain, since a ≤ b ≤ c ≤ d ≤ e ≤ f ≤ g e d c b a Partial order

g In a partial order, a chain is a subset that is totally ordered. f {a, b, c, d, e, f, g} is a chain, since a ≤ b ≤ c ≤ d ≤ e ≤ f ≤ g e x d {b, d, f} is a chain, since b ≤ d ≤ f c b a Partial order

g In a partial order, a chain is a subset that is totally ordered. f {a, b, c, d, e, f, g} is a chain, since a ≤ b ≤ c ≤ d ≤ e ≤ f ≤ g e x d {b, d, f} is a chain, since b ≤ d ≤ f c BUT {x, d, f} is NOT a chain, since x is not comparable with d or f. b a Partial order

Lattice Chains in 2-D The x-coordinates form a non-decreasing sequence
between 0 and 6 The y-coordinates form a non-decreasing sequence between 0 and 4 What are we counting!? Well TWO things… Delannoy paths, and lattice chains. Let’s look at Delannoy paths in 2D. Begin w/ Rectangular grid…. This is our Integer lattice. In this case, 6 by 4. Want to walk from point A to point B. 2 1 2 3 6 4 Length 4

Lattice Chains in 2-D 2 1 2 3 6 4 Length 4 4 1
What are we counting!? Well TWO things… Delannoy paths, and lattice chains. Let’s look at Delannoy paths in 2D. Begin w/ Rectangular grid…. This is our Integer lattice. In this case, 6 by 4. Want to walk from point A to point B. 2 1 2 3 6 4 Length 4 4 1

Lattice Chains in 2-D 2 1 2 3 6 4 Length 4 2 1 2 4 1
What are we counting!? Well TWO things… Delannoy paths, and lattice chains. Let’s look at Delannoy paths in 2D. Begin w/ Rectangular grid…. This is our Integer lattice. In this case, 6 by 4. Want to walk from point A to point B. 2 1 2 3 6 4 Length 4 2 1 2 4 1

Lattice Chains in 2-D What are we counting!? Well TWO things… Delannoy paths, and lattice chains. Let’s look at Delannoy paths in 2D. Begin w/ Rectangular grid…. This is our Integer lattice. In this case, 6 by 4. Want to walk from point A to point B.

How many lattice chains?
We count the # of chains that omit the last point (6,4)… and then double that. Let C(6,4) = # of lattice chains in L(6,4)

# of lattice chains in L(6,4)

+ C(6,3)

+ C(6,3) - C(5,3)

+ C(6,3) - Inclusion/ Exclusion C(5,3)

+ C(6,3) INCLUSION EXCLUSION!! C(6,4) = 2·[C(5,4) + C(6,3) - C(5,3)] RECURRENCE - C(5,3)

1 2 3 4 5 6 8 16 32 64 128 12 80 192 448 1024 104 304 832 2176 5504 1008 3072 8832 24320 10272 32064 95104 107712 341504

1 2 3 4 5 6 8 16 32 64 128 12 80 192 448 1024 104 304 832 2176 5504 1008 3072 8832 24320 10272 32064 95104 107712 341504 …times 2!

1 2 3 4 5 6 2·1 22·1 23·1 24·1 25·1 26·1 27·1 22·3 23·4 24·5 25·6 26·7 27·8 23·13 24·19 25·26 26·34 27·43 24·63 25·96 26·138 27·190 25·321 26·501 27·743 26·1683 27·2668 27·8989

1 2 3 4 5 6 7 9 11 13 25 41 61 85 63 129 231 377 321 681 1289 1683 3653 8989 1 2 3 4 5 6 2·1 22·1 23·1 24·1 25·1 26·1 27·1 22·3 23·4 24·5 25·6 26·7 27·8 23·13 24·19 25·26 26·34 27·43 24·63 25·96 26·138 27·190 25·321 26·501 27·743 26·1683 27·2668 27·8989

1 2 3 4 5 6 7 9 11 13 25 41 61 85 63 129 231 377 321 681 1289 1683 3653 8989 Magic on the Diagonal! 1 2 3 4 5 6 2·1 22·1 23·1 24·1 25·1 26·1 27·1 22·3 23·4 24·5 25·6 26·7 27·8 23·13 24·19 25·26 26·34 27·43 24·63 25·96 26·138 27·190 25·321 26·501 27·743 26·1683 27·2668 27·8989

Magic on the Diagonal! mystery elsewhere 1 2 3 4 5 6 7 9 11 13 25 41
1 2 3 4 5 6 7 9 11 13 25 41 61 85 63 129 231 377 321 681 1289 1683 3653 8989 Magic on the Diagonal! 1 2 3 4 5 6 2·1 22·1 23·1 24·1 25·1 26·1 27·1 22·3 23·4 24·5 25·6 26·7 27·8 23·13 24·19 25·26 26·34 27·43 24·63 25·96 26·138 27·190 25·321 26·501 27·743 26·1683 27·2668 27·8989 mystery elsewhere

? ? ? ? Magic on the Diagonal! mystery elsewhere 1 2 3 4 5 6 7 9 11 13
1 2 3 4 5 6 7 9 11 13 25 41 61 85 63 129 231 377 321 681 1289 1683 3653 8989 ? Magic on the Diagonal! ? 1 2 3 4 5 6 2·1 22·1 23·1 24·1 25·1 26·1 27·1 22·3 23·4 24·5 25·6 26·7 27·8 23·13 24·19 25·26 26·34 27·43 24·63 25·96 26·138 27·190 25·321 26·501 27·743 26·1683 27·2668 27·8989 ? mystery elsewhere ?

Stanley’s 2D Theorem …where… C(n,n) = # chains in L(n,n) and
D(n,n) = # Delannoy paths in L(n,n).

History of this Result This 2D case was first proved by Stanley in his text Enumerative Combinatorics (using generating functions). He posed the challenge to give a combinatorial proof. Sulanke came up with a bijective proof (elaborate!), and attempts at finding D(n,n) and C(n,n) appear sparsely strewn in a wide variety of contexts.

Crash Course in Gen Functions
You want to find a sequence, and you know: a0 = 3, and ai = 2ai-1 for i >0. Let F(x) = a0 + a1 x + a2 x2 + a3 x3 + … So 2x F(x) = 0 + 2a0 x + 2a1 x2 + 2a2 x3 + … Therefore 2x F(x) = F(x) - 3 So So ai = 3(2i)

Crash Course in Gen Functions, II
We have a recurrence: D(0,i) = D(i,0) = 1, and D(i,j) = D(i,j-1) + D(i-1,j) + D(i-1,j-1) Let So

Crash Course in Gen Functions, III
We have two gen functions we can’t expand: and By a technique of Stanley, diff eqs for each can be found convert to diff eqs for gen funcs of ‘diagonals’ these diff eqs can be solved, results compared LIMITATIONS…. No explicit formula No information about the off-diagonal terms Computationally this works in 2 dimensions, chokes in 3 but CPR can get it limping through, and is positively hopeless in 4 or more dimensions.

But numerical evidence clearly suggested the result holds in
1 2 3 4 5 6 7 9 11 13 25 41 61 85 63 129 231 377 321 681 1289 1683 3653 8989 But numerical evidence clearly suggested the result holds in any dimension. 1 2 3 4 5 6 2·1 22·1 23·1 24·1 25·1 26·1 27·1 22·3 23·4 24·5 25·6 26·7 27·8 23·13 24·19 25·26 26·34 27·43 24·63 25·96 26·138 27·190 25·321 26·501 27·743 26·1683 27·2668 27·8989 No explicit formula No info about off-diagonal Computationally this: works in 2 dimensions, chokes in 3 but CPR gets it limping through, positively hopeless in 4 or more dimensions.

Delannoy paths (higher dimensions)

z x

(5,3,2) y z (0,0,0) x

5 3 2

1

1 1

1 1 1 2

1 1 1 2 2 1

1 1 1 2 2 1 3 2

1 1 1 2 2 1 3 2 4 2

1 1 1 2 2 1 3 2 4 2 5 3 2

1 1 1 2 2 1 3 2 4 2 5 3 2 Length 8

1 1 2 1 3 1 3 1 3 2 1 4 2 1 4 3 1 4 3 2 5 3 2 Length 11

Another path of maximum length Length 11

A path of minimum length Length 6

Lattice Chains (higher dimensions)

Chains (higher dimensions)
1 2 1 4 2 1 5 3 2 Length 5

Combinatorial Approach?

1 2 3 4 5 6 7 9 11 13 25 41 61 85 63 129 231 377 321 681 1289 1683 3653 8989 An Explicit Formula!

1 1 2 1 3 1 4 2 4 3 5 4 6 4 8 steps

1 1 2 1 3 1 4 2 4 3 5 4 6 4 1 8 steps

1 1 2 1 3 1 4 2 4 3 5 4 6 4 1 1 1 1 1 1 1 1 8 steps

Delannoy paths in L(6,4) (6,4) (0,0) Length 9 6 4 1 1 1 1 1 1 1 1
6 4 1 1 1 1 1 1 1 1 8 steps

Delannoy paths in L(6,4) Inclusion/ Exclusion Length 9 6 4 1 1 1 1 1 1
6 4 1 1 1 1 1 1 1 1 8 steps # ways to distribute six 1’s into 8 boxes # ways to distribute four 1’s into 8 boxes …TIMES… Inclusion/ Exclusion BUT … no simultaneous 0’s! (the empty step is not allowed)

Delannoy paths in L(6,4) Length 9 6 4 1 1 1 1 1 1 1 1 8 steps
6 4 1 1 1 1 1 1 1 1 8 steps # ways to distribute six 1’s into 8 boxes # ways to distribute four 1’s into 8 boxes …TIMES… BUT … no simultaneous 0’s! (the empty step is not allowed)

Delannoy paths in L(6,4) Length 9 6 4 1 1 1 1 1 1 1 1 8 steps
6 4 1 1 1 1 1 1 1 1 8 steps # ways to distribute six 1’s into 8 boxes # ways to distribute four 1’s into 8 boxes …TIMES… BUT … no simultaneous 0’s! (the empty step is not allowed) …and SUMMED to account for paths with different # of steps!

1 2 3 4 5 6 7 9 11 13 25 41 61 85 63 129 231 377 321 681 1289 1683 3653 8989 An Explicit Formula!

An Explicit Formula! General Form…
1 2 3 4 5 6 7 9 11 13 25 41 61 85 63 129 231 377 321 681 1289 1683 3653 8989 An Explicit Formula! General Form… Alternating Sum of Products of Binomial Coefficients.

Similar arguments generalize to give an explicit formula in any dimension. The recurrence relations and generating function approaches generalize as well.

The Main Result! So the number of Delannoy paths is given by
We were also able to show the number of chains is given by

1 2 3 4 5 6 7 9 11 13 25 41 61 85 63 129 231 377 321 681 1289 1683 3653 8989 1 2 3 4 5 6 2·1 22·1 23·1 24·1 25·1 26·1 27·1 22·3 23·4 24·5 25·6 26·7 27·8 23·13 24·19 25·26 26·34 27·43 24·63 25·96 26·138 27·190 25·321 26·501 27·743 26·1683 27·2668 27·8989

This is our Main Result! So the number of Delannoy paths is given by
We show the number of chains is given by We aren’t looking for ‘a’ formula for C, we need ‘this’ formula for C. Notice its form. Power of 2 times something. Suggests ‘a subset’ paired with ‘something’. That ‘something’ is a ‘REDUCIBLE chain’. The # of pairings of a subset with a ‘reducible chain’.

Reducible vs. Non-reducible
Consider the following two chains in L(3,3,3): Truncating the last coordinates, we get: Reducible k-chains truncate to k-chains.

y z x

Counting Reducible Chains is “pretty easy”

Making Chains Reducible

Going Backward A = { 0,1, 2 }

So the number of chains is: a power of two (counting subsets) times
something else (# of reducible chains).

So these 2 formulas give the diagonal result as a corollary!
But the way I’ve described it is a LIE. It’s how I thought about it 5 years ago. This approach didn’t work, because it is not really a bijection. But if you forget about something and pick it up again later enough times, you’ll eventually see things you didn’t at first.

Corollary …where… C(n,…,n) = # chains in L(n,…,n) and
D(n,…,n) = # Delannoy paths in L(n,…,n).

Bibliography Banderier and Schwer. Why Delannoy numbers? Journal of Statistical Planning and Inference, 135(1):4054, 2005. Bertoin and Pitman. Path transformations connecting Brownian motion, excursion and meander. Bulletin des Sciences Mathématiques, 118(2):147166, 1994. Caughman, Haithcock, Veerman. A note on lattice chains and Delannoy numbers. Discrete Math., 308: , 2008. Duchi and Sulanke. The 2n+1 factor for multi-dimensional lattice paths with diagonal steps. Séminaire Lotharingien de Combinatoire, 51, Article B51c. Hetyei. Central Delannoy numbers and balanced Cohen-Macaulay complexes. Annals of Combinatorics, 10: , 2006. Nelson and Schmidt. Chains in power sets. Mathematics Magazine, 64:23-31,1991. Stanley. Enumerative Combinatorics, Volume 2. Cambridge University Press, Cambridge,1999.

THANK YOU!

Counting Lattice Chains and Delannoy Numbers

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