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•i LATTICE POINT SOLUTION OF THE GENERALIZED PROBLEM OF TERQUEi
. AND AN EXTENSION OF FIBONACCI NUMBERS .
C . A . CHURCH, J r . and H. W . G O U L D , W . Virginia University, Morgantown, W . V a .
In this paper we give a simple lattice point solution to a generalized permutation problem of Terquem and develop some elementary results for the
extended Fibonacci numbers associated with the permutation problems
The classical permutation problem of Terquem [12] has been stated by
Riordan [10, p„ 17, ex„ 15] in the following manner,.
Consider combinations
of n numbered things in natural (rising) order, with f (n,r) the number of r combinations with odd elements in odd position and even elements in even positions, or, what is equivalent, with f (n,r) the number of combinations with an
equal number of odd and even elements for r even and with the number of odd
elements one greater than the number of even for r odd.
It is easy to show that f(n,r) = f (n - 1, r - 1) + f(n - 2, r), with f(n, 0)
= 1, and explicitly
(1)
f(n,r) =
Moreover,
n
(2)
f (n) = ^T f (n, r) = f (n - 1) + f (n - 2)
r=o
so that f(n) is an ordinary Fibonacci number with f(0) = 1 and f(l) = 2„
A detailed discussion of Terquem f s problem is given by Netto [8, pp. 8487] and Thoralf Skolem [8, pp. 313-314] has appended notes on an extension of
the problem in which the even and odd question is replaced by the more general
question of what happens when one uses a modulus m to determine the position
of an element in the permutation.
"^Research supported by National Science Foundation Grant GP-482.
59
60
LATTICE POINT SOLUTION OF THE GENERALIZED
M o r e p r e c i s e l y , for a modulus m > 2,
stated a s follows.
[Feb.
Skolem T s g e n e r a l i z a t i o n m a y b e
F r o m among the f i r s t n n a t u r a l n u m b e r s let f(n,r;m)
de-
note the n u m b e r of combinations in n a t u r a l o r d e r of r of t h e s e n u m b e r s such
that the j
(3)
element in t h e combination i s congruent to j modulo m. That i s ,
f(n,r;m) = N < a 1 a 2 o e ° a r :
1 < a* < a 2 < * " ° < a r < n,
a. = j (mod m)>
Z
1< B,t< a 2 <° • • < a r < n
a. = j (mod m)
C o n s i d e r the a r r a y in F i g . 1, w h e r e the l a s t e n t r y is
r + km,
with
k =
m
s i n c e r + k m < n i m p l i e s that t h e l a r g e s t i n t e g r a l v a l u e of k cannot exceed
(n - r ) / m 0
T h i s a r r a y contains t h o s e , and only t h o s e , e l e m e n t s from among
1, 2, • • • , n which m a y a p p e a r in a combination,,
s i s t s of all t h o s e e l e m e n t s < n
m a y a p p e a r in the j
-
column con-
in t h e s a m e congruence c l a s s (mod m) which
position.
(0,0)
1 —
#-$*» X
1 +m• l + 2m
i-
That i s , t h e j
1 + km
i.
•2 + m -
-3 + m -
-2 + 2m
-3 + 2 m
-2 + k m
i.
• 3 + km
•r +m •
2m
-i.
-r + km —#
B = (r,s)
Fig 0 1
F r o m t h e l a t t i c e appended to the a r r a y In Fig„ 1, we can s y s t e m a t i c a l l y
w r i t e out t h e d e s i r e d combinations, and evaluate f(n, r;m)„
To get the d e s i r e d r e s u l t , let " a path from A to BfT m e a n a path along
t h e v e r t i c a l and h o r i z o n t a l s e g m e n t s of t h e l a t t i c e , always moving downward o r
from left to right (we t a k e the p o s i t i v e x - a x i s to the right, t h e p o s i t i v e y - a x i s
1967]
PROBLEM OF TERQUEM AND AN EXTENSION
61
OF FIBONACCI NUMBERS
downward, thus agreeing with the informal way of writing down the permutations)*
Each such path will generate a combination of the desired type, and conversely,
as follows? Starting at A each horizontal step picks up an entry and vertical
steps line up entries 0
Now, It is well known how many lattice paths there a r e
from a = (0,0) to B = (r,s). MacMahon [7, VoL I, p„ 167] shows that this
number is precisely
(
hi our case s = [(n - r ) / m ] c
)
Thus we have at once that
r +
f (n, r;m)
(4)
r
(m - l)r"
m\[-
HI
r
as found by Skolem0 Terquem ! s (1) follows when m = 2„
To illustrate, we
consider some examples 0
Example 1. Let n = 12, r = 3, m = 40 Then the corresponding array
is
1r——
1
E
c
«—
• •
—
2——i
1
3
7
fiD
1
3 _ J ^-»-10—*-
and the ten combinations are
1 2
3
1 2
7
1
2
11
1 6
1
6
7
11
1
11
10
5
6
7
5
5
6
10
11
11
9
10
11
and the particular combination 5, 10, 11 corresponds to the path indicated by
arrows. Informally, one writes out the combinations by paths from the left
column to the right column, moving horizontally and/or diagonally,,
The clue
to a systematic count is found by superimposing the rectangular grid*
Example 2„ Let n = 12, r = 4, m = 30 Then the corresponding array
is
62
LATTTCE POINT SOLUTION OF THE GENERALIZED PROBLEM [Feb*
2
1
FH
1|
3
—»-5—*-
1
11
n
1
*7
t
o
*
8
4
11
». Q — , JL^^io—^-
and the fifteen combinations a r e
1 2
1 2
1
2
1 2
1
2
3
3
4
7
3
1 2 9
1 5
6
10
7
4
4
5 6
5 6
7
10
10
1
5
6
10
4
5
9
10
6
7
1
5
9
10
4
8 9
10
6
10
1
8
9
10
7
8 9
10
and the combination 4, 5, 9, 10 corresponds to the path indicated by arrows,
It is felt that our proof shows atruism of mathematics: one may often find
a simpler proof by embedding a given problem (Terquem f s) in a more general
setting., The lattice point enumeration we used is well known, but may not be
apparent in the original problem because of its specialized form 0
The extended Fibonacci numbers, in analogy to (2), are now defined by
n
(5)
f(n) = f m (n) =
£
j*n + (m - l)r"
m
r=o satisfy the recurrence relation
and it is not difficult to verify that they
(6)
f m (n) - f m (n - 1) + f m (n - m)
For example, with m = 3 we have the sequence 1, 2, 3, 4, 6, 9,13,19, 28,
By well-known theorems in the theory of linear difference equations, if the distinct roots, of the equation
(7)
t m - t111"1 - i - o
a r e ti, t 2 , • • • t m * then there exist constants C r such that
1967] O F TERQUEM AND AN EXTENSION OF FIBONACCI NUMBERS
63
HI
(8)
f
m
(n) = \ ^ C
ZmJ
r
tn
rr
This generalizes the familiar formulas
-o
a - b
F
= -~
• ,
n
a - b
,
L
= a
n , ,n
+ b
n
for the F i b o n a c c i - L u c a s n u m b e r s . T h e c o n s t a n t s C
,
*
m a y b e d e t e r m i n e d from
t h e s y s t e m of m l i n e a r equations in C :
m
(9)
7 J C r t 3 r = j + 1,-
for
j = 0, 1, 2, •••, m - 1 .
r=i
F o r example 9 when m = 3, an a p p r o x i m a t e solution of the equation (7) i s given
by
w h e r e i2 = ' - 1 .
I
t j = 1.4655 ,
t2 -
-0.23275 + 0 o 792551 ,
t 3 = -0 o 23275 - 0 o 792551 ,
Relations (5) through (9) a r e given by Skolem [ 8 , 313-314]*
When m = 3 t h e exact solution of (7) i s given by
ti = A + B + -|
(11)
t2 = J - A + i
1
where
A + B
+
A^B
^
A - B ./-o
p
64
LATTICE POINT SOLUTIONS OF THE GENERALIZED PROBLEM [ F e b .
A s a p a r t i a l check on t h e v a l u e s of t h e r o o t s , we note t h e following t h e o r e m
from t h e t h e o r y of equations.
Let
m
n (t - t.) = t m - t m
(12)
- z
Then
m l
(13)
lL ]
j=i
=
m
£ \b>
X
" m) ^ '
n> 15
k=o
where
A
A
/ i.\
(a b)
k '
a
/ a + bk \
= rrbk (
k
)•
T h i s may b e c o m p a r e d with the well-known [ 2 , 3, 4, 5 ]
xa = ^ T Ak(a,b)zk
(14)
,
with z =
,
k=o
expansion
x - 1
b
x
which w a s actually found by L a g r a n g e in his g r e a t m e m o i r of 1770 (Vol, 24 of
P r o c . of t h e B e r l i n Academy of Sciences) and which l e a d s at once to the g e n e r a l
addition t h e o r e m d i s c u s s e d in [ 2 , 3, 4, 5 ]
See r e l a t i o n (20), this paper,,
m
m—l
F o r the equation t - t
- z = 0,
roots t .
(15)
a s f i r s t noted by H. A. Rothe„
we define t h e power s u m s of t h e
by
S(n) -
£l)
1967] OF TERQUEM AND AN EXTENSION OF FIBONACCI NUMBERS
Since t . "
3
65
+ z = t . , we find that
3
m
{t*" 1 + z t ^ 1 1 1 J = £
S(n - 1) + zS(n - m) = £
j=i
t ^ / t ^
j=i
1
+ z\
^
- J ]
/
t^t*1,
j=i
so that S(n) itself a l s o s a t i s f i e s a F i b o n a c c i - t y p e r e c u r r e n c e
(16)
S(n) = S(n - 1) + zS(n - m) .
Using t h e v a l u e s z = 1, m = 3, t h e p r e v i o u s r o o t s (10) yield t h e a p p r o x i m a t e
v a l u e s (by log t a b l e s ) : S(l) - 1,
very nearlyc
S(2) = 0 e 9998, S(3) - 3,9995, and S(4) = 5
T h i s gives a p a r t i a l check on (10)o
In any event 5 we m a y c o n s i d e r t h e s e q u e n c e defined by (13), (15), (16) a s
a kind of extended Fibonacci sequence,,
(17)
S(n)
™
n
n - (m - l ) k ( •
Z^
k=o
hi p a r t i c u l a r ,
|- (m
k
l)k
n > 1 ,
s a t i s f i e s (16) j u s t a s (5) s a t i s f i e s (6). T h e r e a r e s i m i l a r i t i e s and c o n t r a s t s if
w e c o m p a r e (17) and (5)0
We a l s o c a l l attention to a n o t h e r such r e s u l t given
r e c e n t l y by J„ A9 Raab [Vj, who found that the sequence defined by
(18)
x
=
[r+ij
\-^
/ n - rk\
n- k(r+i)
satisfies
(19)
x
n
= ax
n - lJ
+ bx
n-r-i
bk
6
^
LATTICE POINT SOLUTIONS OF THE GENERALIZED PROBLEM [Feb„
Formula (13) is substantially that given by Arthur Cayley [!]<,
The classical
Lagrange inversion formula for series is inherent in all these formulas,,
One
should also compare the Fibonacci-type relations here with the expansions
given in [5]„
For m = 3, (17) gives the sequence 1,1, 4, 55 6,10,15, 21, 31,
We als.o call attention to the two well-known special cases
, ,
£ (n i k) r -2kzk
n+i
__ x
=
n+i
- y
x - y
k=o
and
V^ / n - k \
n
k n k
=r \ / -
k=o
n-2k-i k
z
=
n +, y n
x11
•
+
'
y
x + y
where x = 1 + Vz + 1, y = 1 -v/z + 1° F and L occur when z = 49
J
v
n
n
Relations (17) and (5) differ because the initial conditions differ., For
z = 1,
(17) satisfies precisely the same recurrence as (5)c
If the initial
values were the same then we would have found a formula for the permutation
problem not unlike (17)„ There are many papers (too numerous to mention) in
which complicated binomial sums are found by lattice point enumerations,, The
convolutions in £2, 3,4, 5J may mostly be found by such counting methods,, We
also note the recent papers of Greenwood |jf] and Stocks [ll] wherein the
Fibonacci numbers occur 0
The convolution addition theorem [2, 3,4, 5] of H Ae Rothe (1793)
(20)
^ A k ( a , b ) A n _ k ( c s b ) - A n (a + c,b) ,
k=o
valid for all real or complex a, b, c (being a polynomial identity in these), has
been derived several times by lattice point methods e We mention only a novel
1967] OF TERQUEM AND AN EXTENSION OF FIBONACCI NUMBERS
67
derivation by Lyness [13j 0 Relation (20) has been rediscovered dozens of times
since 1793, and its application in probability., graph theory, analysis, and the
enumeration of flexagons, etc„ , shows that the theorem is very usefuL In fact,
it is a natural source of binomial identities,, We should like to raise the question here as to whether any analogous relation involving the generalized Terquem
coefficients (4) existSo It seems appropriate to study the generating function
defined by
T(x;a,b) = J^
(21)
(
fa + (b - l ) n 1 \
n=o
for as general a and b as possible., If b is a natural number and a is an
integer >0, the series terminates with that term where n = a, as is evident
from the fact that a + (b - l)n < bn for n > a and the fact that |
1 = 0 for
k < n when n > 0, provided k > 0o We also note that for arbitrary complex
a and | x | < 1
T(x;a,l) =
2 (») *" - {1+ x)"
n=o * '
so that in this case we do have an addition theorem:
T(x;a 9 l)T(x;c 5 l) = T(x ; a + c , l ) .
This, of course, corresponds to the case b = 0 in formula (20); the relation
implies the familiar Vandermonde convolution or addition theorem.
There does not seem to be any especially simple closed sum for the series
n
(22)
C n (a,c,b) -
which occurs in
^
k=o
/ f a + ( b - l ) k l \ / f c + (b - l')(n - k)] \
68
LATTICE POINT SOLUTION OF THE GENERALIZED PROBLEM
OF TERQUEM AND AN EXTENSION OF FIBONACCI NUMBERS
T(x;a,b)T(x,c,b) = V
xnCn(asc?b)
Feb„
,
n=o
for a r b i t r a r y b 6
REFERENCES
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A„ Cayley, "On the Sums of C e r t a i n S e r i e s A r i s i n g f r o m the
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Equation
2.
Ho Wo Gould, "Some G e n e r a l i z a t i o n s of Vandermond 1 s Convolution," A m e r .
Math, Monthly, 63(1956), 8 4 - 9 1 .
3e
H0 Wo Gould, " F i n a l A n a l y s i s of Vandermonde T s
Math. Monthly, 64(1957), 409-415.
4C
Ho W„ Gould, " G e n e r a l i z a t i o n of a T h e o r e m of J e n s e n Concerning Convolut i o n s , " D u ^ e M a t h o J o , 27(1960), 71-76 c
5.
Ho W0 Gould, "A New Convolution F o r m u l a and Some New Orthogonal
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Chelsea R e p r i n t , Ne Y. 1960 o
8.
E. Netto, Lehrbuch d e r Combinatorik,
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Convolution,"
Leipzig, 1927, 2nd E d . ,
Amer.
Fibonacci
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9„ J . A. R a a b , "A G e n e r a l i z a t i o n of the Connection Between the F i b o n a c c i
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3, pp 0 2 1 - 3 1 .
10o
J . R i o r d a n , An Introduction to Combinatorial A n a l y s i s , N0 Y», 1958.
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D0 Ro Stocks, J r c , "Concerning L a t t i c e P a t h s and F i b o n a c c i N u m b e r s , "
F i b o n a c c i Q u a r t e r l y , VoL 3, 1965, pp c 143-145 0
12o
Oo T e r q u e m , "Sur un symbole c o m b i n a t o i r e d f E u l e r et son Utilite dans
P A n a l y s e , " J 0 Math 0 P u r e s Appl. , 4(1839), 177-184 0
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Ro Co L y n e s s , "Al Capone and t h e Death R a y , " Matho G a z e t t e ,
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* *** *
25(1941),