Perfect matchings for the three-term Gale-Robinson sequences ppsx

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Perfect matchings for the three-term
Gale-Robinson sequences

Mireille Bousquet-Mélou (LaBRI), James Propp, Julian West
(Submitted on 17 Jun 2009)

∗
†
∗
†
the electronic journal of combinatorics 16 (2009), #R125 1
s(n) = (s(n − 1)s(n − 3) + s(n − 2)
2
)/s(n − 4),
s(n)s(n − 4) = s(n − 1)s(n − 3) + s(n − 2)
2

,
s(n)s(n − k) = s(n − 1)s(n − k +1)+s(n−2)s(n−k + 2)+· · ·+ s(n − ⌊k/2⌋)s(n − ⌈k/2⌉).
s(0) = s(1) = · · · = s(k − 1) = 1
k
k = 4, 5 , 6, 7
a(n)a(n − m) = a(n − i)a(n − j) + a(n − k)a(n − ℓ),
a(0) = a(1) = · · · = a(m−1) = 1 m = i +j = k + ℓ
(i, j, k, ℓ) (3, 1, 2, 2) (4, 1, 3, 2)
i, j, k, ℓ > 0 i + j = k + ℓ = m
a(0), a(1), . . .
a(n)a(n − m) = a(n − g)a(n − h) + a(n −
i)a(n − j) + a(n − k)a(n − ℓ) g, h, i, j, k, ℓ, m
the electronic journal of combinatorics 16 (2009), #R125 2
P (n; i, j, k, ℓ) n  0 n
a(n)
i = j = k = ℓ = 1
P (25; 6, 2, 5, 3) a(25) a(n)
(i, j, k, ℓ) = (6, 2, 5, 3)
P (n; i, j, k, ℓ)
P (n
′
; i, j, k, ℓ) n
′
< n
P (n; i, j, k, ℓ)
a(n)
P (n) ≡ P (n; i, j, k, ℓ) a(0), a(1), a(2),
a(0) =
a(1) = · · · = a(m − 1) = 1
P (n) a(n) n a(n)

a(n
1
)a(n
2
)
p(n) ≡ p(n; w, z)
p(n)p(n − m) = w p(n − i)p(n − j) + z p(n − k)p(n − ℓ),
i+j = k +ℓ = m p(0) = p(1) = · · · = p(m−1) = 1
w z
p(n; u
2
, v
2
) P (n; i, j, k, ℓ)
the electronic journal of combinatorics 16 (2009), #R125 3
u
v p(n)
Z
a(n) A(n, p, q)
A(n, p, q)A(n−m, p, q) = A(n−i, p−1, q)A(n−j, p+1, q)+A(n−k, p, q+1)A(n−ℓ, p, q−1).
(−1, 0), (1, 0), (0, 1), (0, −1)
the electronic journal of combinatorics 16 (2009), #R125 4
A(n, p, q) = x
n,p,q
n m − 1 p, q
x
n,p,q
A(n, p, q) n  m
n, p, q, r, s A(n, p, q) A(n, r, s)
A(n, p, q)

x
n,r,s
A(n, p, q)
Z[x
±1
n,r,s
]
x
n,r,s
A(n, p, q) a(n)
1
i = j = k = ℓ = 1 A(n, p, q)
A(n, p, q)
A(n, p, q)
n
the electronic journal of combinatorics 16 (2009), #R125 5
G
(V, E) V E
V v E
v G H = (V
′
, E
′
) V
′
⊂ V E
′
⊂ E
V
′

= V H G
G = (V, E) H = (V
′
, E
′
) G ∩ H = (V ∩ V
′
, E ∩ E
′
)
G ∪ H = (V ∪ V
′
, E ∪ E
′
) G = (V, E)
H = (V
′
, E
′
) G \ H (V
′′
, E
′′
) V
′′
= V \ V
′
E
′′
E \ E

′
V
′′
G = (V, E) E
′
E
V E
′
G
m(G) G
E
E
′
G
M(G) :=

E
′

e∈E
′
e,
E
′
G e
n
e
n
e
x y e = {x, y} E

x y {x, y} E n
e
G
e n
e
= 1
2k − 1 1, 3, . . . , 2k − 3, 2k −
1, 2k−3, . . . , 3 , 1
A
the electronic journal of combinatorics 16 (2009), #R125 6
9
e
s
n
w
9
2k − 1 A
N
2k − 3
A
2k − 3 A A
N
A
A A
S
, A
W
A
E
A

C
A 2k − 5
A
M(A)M(A
C
) = nsM(A
W
)M(A
E
) + ewM(A
N
)M(A
S
),
n, s, w, e
A
a(n) n  2
2n − 3
a(n)a(n − 2) = 2a(n − 1)
2
n  2 a(0) = a(1) = 1
a(n) i = j = k = ℓ = 1
a(n) = 2
(
n
2
)
A G A
n, s, w e G
N

= G ∩ A
N
G
S
, G
W
, G
E
G
C
M(G)M(G
C
) = nsM(G
W
)M(G
E
) + ewM(G
N
)M(G
S
).
the electronic journal of combinatorics 16 (2009), #R125 7
A
E
A
W
A
C
A
N

A
S
9
G A G
A M(G)
a = 0 M(A) a A G
M(G
N
) M(A
N
)
a = 0 a A G
2k − 1 P = (V, E)
a
1. i + j + 1
(0, 1), (1, 2) . . . , (i−1, i) (0, 0), (1, −1), . . . , (j, −j) i  1 j  0
L
m
m
L
−j
< · · · < L
−1
< L
0
= 2k − 1 = L
1
> L
2
> · · · > L

i
.
2. V
the electronic journal of combinatorics 16 (2009), #R125 8
3. e = {(a, b), (a, b + 1)}
V a + b e E e
P e E a e
(0, 0) ( 0 , 1)
(a, b) (a, b + 1) a + b
b. c.a.
(0, 0)
15 a
b c
b c a
c
1
b
(0, 0) (1, 1)
R
2
P
P 2k − 1
2k −1 ℓ r
P P (ℓ, r)
P ℓ (0, 0)
0 1 −1
P
P
the electronic journal of combinatorics 16 (2009), #R125 9
S

y = x > 0 −y = x > 0
S
S −j i m
m m + 1 m > 0 m < 0
m m−1 m+1
S
P P
P v P
P v u u
P
¯
P P
P
P P
¯
P
P
P
¯
P
P
P
P P
¯
P P
(a, b)
the electronic journal of combinatorics 16 (2009), #R125 10
P
0
P (−a, −b)

P P
0
(a, b)
P b
0
P b
1
P b
0
b
2
P b
1
b
−1
−1
P b
0
b
m
P
b
k
˜
P
Q P
˜
P
Q b
0

Q b
1
Q
˜
P
P
P
¯
P = P
P
¯
P P \
¯
P
H
¯
P
P H m(P )  m(
¯
P )
P
¯
P
P
¯
P m(
¯
P ) = m(P)
Q P
Q P

¯
P Q
Q v = (a, b)
1 Q v Q
the electronic journal of combinatorics 16 (2009), #R125 11
v Q b > 1
u
v u v Q
u
Q Q
′
Q u v
u v Q Q
′
v
1
= (a − 1, b + 1) Q v
1
Q
′
i
v
j
= (a − j, b + j) Q 0  j  i
v = v
0
, v
1
, . . . , v
i

Q
∗
b = 1 Q
(a − j, 1 + j) (a − j, −j) Q
∗
b
v
v
2
v
1
v
v
2
v
1
v
v
Q
Q
Q
∗
Q
∗
m(Q) = m(Q
∗
) Q
∗
¯
P

Q Q
∗
¯
P
¯
P
¯
P
m(
¯
P ) = m(P) Q = P
Q
∗
Q
∗∗
m(P ) = m(Q) = m(Q
∗
) =
m(Q
∗∗
) = · · ·
¯
P =
¯
Q = Q
∗
= Q
∗∗
= · · ·
P

¯
P
¯
P
m(Q) m(
¯
P ) = m(P)
the electronic journal of combinatorics 16 (2009), #R125 12
P 2n + 1 A
P A 2n + 1
G A P
A A G
A
P G P
G
P
G
P
G
N
= G ∩ A
N
G
N
, G
S
, G
W
, G
E

G
C
P
N
P
S
P
W
P
E
P
C
P
N
P A
N
P
N
P
S
P
W
P
E
P
C
P P
ℓ
0
r

0
R
0
P r
1
r
−1
R
0
r
0
, r
1
r
−1
P ℓ
′
0
R
0
ℓ
0
r
′
0
R
0
r
0
P

P P
N
P
S
P r
1
r
−1
P
W
P
E
r
′
0
ℓ
′
0
P
C
(ℓ
′
0
, r
′
0
)
P
P
N

P
S
P
E
P
W
P
the electronic journal of combinatorics 16 (2009), #R125 13
P
C
P
N
P
E
P
W
P
S
r
−1
ℓ
0
r
1
r
0
r
′
0
ℓ

′
0
P
P
N
r P r > 0 r − 1 P
N
r < 0 r P r + 1 P
S
P 2 1
P
E
P
W
G P
A P G m(G) = m(P )
G
N
P
N
P
m(P )m(P
C
) = m(P
W
)m(P
E
) + m(P
N
)m(P

S
).
2n − 3 n
P
W
the electronic journal of combinatorics 16 (2009), #R125 14
a(n)a(n − 2) = a(n − 1)a(n − 1) + a(n − 1)a(n − 1),
a(0) = a(1) = 1
a(n)a(n − m) = a(n − i)a(n − j) + a(n − k)a(n − ℓ),
a(n) = 1 n = 0, 1, . . . , m − 1 i, j, k ℓ
i + j = k + ℓ = m
j = min{i, j, k, ℓ}.
(P (n))
n0
≡
(P (n; i, j, k, ℓ))
n0
{i, j, k, ℓ} i+j = k+ℓ = m
• P (n)
C
P (n − m) (2, 0)
• P (n)
W
P (n − i)
• P (n)
E
P (n − j) (2, 0)
• P (n)
N
P (n − k) (1, 1)

• P (n)
S
P (n − ℓ) (1, −1)
{(a, b), (a, b + 1)}
a + b U L
P (n; i, j, k, ℓ) r  0
U(n, r, c) = 2c + r − 3 − 2

mc + kr + i − n − 1
j

,
L(n, r, c) = 2c + r − 3 − 2

mc + ℓr + i − n − 1
j

.
k ℓ U(n, 0 , c) = L(n, 0, c)
U L
0
(0, 0)
r  0 U(n, r, c)
c = 0, 1, . . . m  2j
the electronic journal of combinatorics 16 (2009), #R125 15
i + j = m j  i U(n, r, c) r
r U(n, r, c) (U(n, r, c), r)
(U(n, r, c), r+1 ) U(n, r, c)+r
U(n, r, 0) < r U(n, r, 0)−r r j  k
r

r U(n, r, 0)
r+1 U(n, r, 0)−r
U(n, r +1, 0)−(r +1) < U(n, r, 0) −r
U(n, r + 1, 0) − (r + 1)  U(n, r, 0) − r − 2 U(n, r + 1, 0) < U(n, r, 0)
r > 0 r − 1
−r  0 L(n, r, 0) >
L(n, r, 1) > · · · r −r
L(n, r, c) (L(n, r, c), −r ) (L(n, r, c), −r + 1)
L(n, 0, c ) = U(n, 0, c) 0
U L
U L
(r, r + 1) (U(n, r, 0), r + 1) (−r, −r) (L(n, r, 0), −r)
r  0
P (n) U(n, 0, 0) < 0
U(n, 0, 0)  −1 U(n, 0, 0) n < m
m = i + j
(i, j, k, ℓ) = (5, 2, 3, 4) P (12) U
L
U(n, r, c) = 2c + r − 3 − 2

7c + 3r − 8
2

,
L(n, r, c) = 2c + r − 3 − 2

7c + 4r − 8
2

.

0 (5, 0) (1, 0) 1
(4, 1) U(12, 2, 0) = 1 < 2
(2, −1)
−2
P (12)
a(n) (5, 2, 3, 4) a(12) = 14
the electronic journal of combinatorics 16 (2009), #R125 16
(2)(4)(4) (4)
(0, 0)
P (12; 5, 2, 3, 4)
U L
r r + 1
r  0 r +1
r  0 r − 1
n, r c U
L
U(n, r, c + 1) + 1  U(n, r + 1, c)  U(n, r, c) − 1
L(n, r, c + 1) + 1  L(n, r + 1, c)  L(n, r, c) − 1.
U(n, r + 1, c) − U(n, r, c + 1) =
2

mc + kr + i − n − 1 + m
j

− 2

mc + kr + i − n − 1 + k
j

− 1.

mc + kr + i − n − 1 + m
j
−
mc + kr + i − n − 1 + k
j
=
ℓ
j
 1,
U(n, r + 1, c) − U(n, r, c + 1)  2 − 1 = 1.
P (n)
P (n)
W
= P (n − i) P (n)
E
= P (n − j) P (n)
N
= P (n − k) P (n)
S
= P (n − ℓ)
P (n)
C
= P (n−m)
the electronic journal of combinatorics 16 (2009), #R125 17
(i, j, k, ℓ) U L
U(n − i, r, c − 1) = U(n, r, c), L(n − i, r, c − 1) = L(n, r, c),
U(n − j, r, c) = U(n, r, c) − 2, L(n − j, r, c) = L(n, r, c) − 2,
U(n − k, r − 1, c) = U(n, r, c) − 1, L(n − ℓ, r − 1, c) = L(n, r, c) − 1,
U(n − ℓ, r + 1, c − 1) = U(n, r, c) − 1, L(n − k, r + 1, c − 1) = L(n, r, c) − 1,
U(n − m, r, c − 1) = U(n, r, c) − 2, L(n − m, r, c − 1) = L(n, r, c) − 2.

L U k ℓ
U
U(n − ℓ, r + 1, c − 1)
= 2(c − 1) + (r + 1) − 3 − 2

m(c − 1) + k(r + 1) + i − (n − ℓ ) − 1
j

= 2c + r − 4 − 2

mc + kr + i − n − 1 − m + k + ℓ
j

= 2c + r − 4 − 2

mc + kr + i − n − 1
j

m = k + ℓ
= U(n, r, c) − 1.
P (n) ≡ P (n; i, j, k, ℓ)
(i, j, k, ℓ) n  m P (n)
P (n)
W
= P (n − i), P (n)
E
= P (n − j),
P (n)
N
= P (n − k), P (n)

S
= P (n − ℓ),
P (n)
C
= P (n − m).
P (n)
W
= P (n − i) P (n)
W
P (n)
P (n)
W
P (n)
P (n)
W
P (n) c = 1, 2, . . . c = 0, 1, . . .
the electronic journal of combinatorics 16 (2009), #R125 18
r  0 P (n)
W
U(n, r, 1), U(n, r, 2), . . .
r r  0 L U
P (n − i) r  0 P (n − i)
U(n − i, r, 0), U(n − i, r, 1), . . . r
U(n − i, r, c − 1) = U(n, r, c )
r P (n)
W
P (n −i)
r < 0 L
P (n)
W

= P (n − i)
P (n)
N
P (n)
S
P (n)
S
r  0 r P (n)
S
r−1 P (n)
r > 0 r P(n)
S
r − 1 P (n)
r > 0 r P(n)
S
r −1 P (n) P (n)
S
−r r = 1, 2, . . . L(n, r, 0), L(n, r, 1), . . .
r r = 0, 1, 2, . . . U(n, r, 1), U(n, r, 2), . . .
r + 2
P (n − ℓ) (1, −1)
−r r = 1, 2, . . .
L(n−ℓ, r−1, c)+1 c  0 r
P (n)
S
P (n − ℓ)
P (n−ℓ) r r = 0, 1, 2, . . .
U(n − ℓ, r + 1, c) + 1 c  0
r + 2 P (n)
S

P (n − ℓ)
P (n)
S
P (n)
N
P (n)
E
P (n)
C
P (n)
N
P (n)
P (n) (i, j, k, ℓ) 0  n < m
P (n) m  n < m+j
P (n)
n  m + j
P (n − i), P(n − j), P (n − k) P(n − ℓ) 2 1
P (n − j)
P (n − i) P(n − j), P (n − k), P (n − ℓ)
(0, 0) (2, 0) (1, 1) (1, −1) 2 1
(0, 0) P (n−i)
the electronic journal of combinatorics 16 (2009), #R125 19
P (10) =
P (7) =P (5) = P (6) =
P (8) =
P (4) =
P (5)
E
= P (4)
P (6)

E
= P (5) P (7)
W
= P (4), P(7)
E
= P (6)
P (9)
N
= P (7)
P (10)
W
= P (7)
P (8)
N
= P (8)
S
= P (6)
P (9) =
P (n − i)
(1, 0) ( 1 , 1) P (n)
P (n − i)
P (n)
a(n)a(n − 4) = a(n − 3)a(n − 1) + a(n − 2)
2
.
(i, j, k, ℓ) = (3, 1, 2, 2) m = 4
P (n) ≡ P(n; i, j, k, ℓ)
(i, j, k, ℓ) a(n) P (n)
a(n) = 1 n < m n  m a(n)
a(n)a(n − m) = a(n − i)a(n − j) + a(n − k)a(n − ℓ).

P (n) n < m
n  m
P (n)
m(P (n))m(P (n)
C
) = m(P (n)
W
)m(P (n)
E
) + m(P (n)
N
)m(P (n)
S
).
m(P (n)
C
) = m(P (n − m))
m(P (n))m(P (n − m)) = m(P (n − i))m(P (n − j)) + xm(P (n − k))m(P (n − ℓ) ),
a(n)
the electronic journal of combinatorics 16 (2009), #R125 20
i, j, k, ℓ i+j = k+ℓ = m
a(n)a(n − m) = a(n − i)a(n − j) + a(n − k)a(n − ℓ),
a(n) = 1 n < m
(i, j, k, ℓ) (6, 2, 5, 3)
n = 25
25
P (n) r
U(n, r, c) c = 0, 1, 2, 3, . . .
r = 0 : {17, 11, 5, −1, . . .}
r = 1 : {14, 8, 2, −4, . . .}

r = 2 : {9, 3, −3, −9, . . .}
r = 3 : {6, 0, −6 − 12, . . .}
r = 4 : {1, −5, −11, −17, . . .}
c = 0 r = 4 r
L(n, r, c)
r = 0 : {17, 11, 5, −1, . . .}
r = 1 : {16, 10, 4, −4, . . .}
r = 2 : {13, 7, 1, −5, . . .}
r = 3 : {12, 6, 0, −6, . . .}
r = 4 : {9, 3, −3, −9, . . .}
r = 5 : {8, 2, −4, −10, . . .}
r = 6 : {5, −1, −7, −13, . . .}
P (25; 6, 2, 5, 3)
the electronic journal of combinatorics 16 (2009), #R125 21
P (25; 6, 2, 5, 3)
25 (6, 2, 5, 3)
i, j, k, ℓ m i + j = k + ℓ = m
w z p(n) ≡ p(n; w, z) p(n) = 1
n < m n  m
p(n)p(n − m) = w p(n − i)p(n − j) + z p(n − k)p(n − ℓ).
p(n) w z
p(n; 1, 1)
P ≡ P(n; i, j, k, ℓ) p(n; u
2
, v
2
)
the electronic journal of combinatorics 16 (2009), #R125 22
P
M(P )

P (n; i, j, k, ℓ) q(n) ≡ q(n; u, v)
P (n) v
u q(n) = 1 n < m
q(n)q(n − m) = u
2
q(n − i)q(n − j) + v
2
q(n − k)q(n − ℓ)
n  m q(n; u, v) = p( n; u
2
, v
2
)
P A
G n s e w
P G M(G)
M(P )M(G \ P ) P
C
P
W
P
E
P
N
P
S
M(P )M(G \ P )M(P
C
)M(G
C

\ P
C
) = nsM(P
W
)M(G
W
\ P
W
)M(P
E
)M(G
E
\ P
E
)
+ ewM(P
N
)M(G
N
\ P
N
)M(P
S
)M(G
S
\ P
S
).
P G G \ P
M(G \ P )

M(P )M(P
C
) = α M(P
W
)M(P
E
) + βM(P
N
)M(P
S
)
α β
M(G \ P )M( G
C
\P
C
)
P
P
(i, j) i + j
the electronic journal of combinatorics 16 (2009), #R125 23
P P

P
>
0, 1, 2, . . . P 1, 2, . . . P
(P
N
)
