A generalization of the Seidel-Entringer-Arnold method for calculating the
alternating permutation numbers (or secant-tangent numbers) leads to a new
operation on integer sequences, the Boustrophedon transform.Comment: very minor change: corrected typo in author list. June 24 2002:
  correction to a proof; additional reference

Millar, Jessica

Sloane, N. J. A.

Young, Neal E.

English

arXiv

A generalization of the Seidel-Entringer-Arnold method for calculating the alternating per-mutation numbers (or secant-tangent numbers) leads to a new operation on sequences, the boustrophedon transform. This paper was published (in a somewhat different form) in J. Combinatorial Theory, Series A, 76 (1996), pp. 44–54

J. Millar

N. J. A. Sloane

N. E. Young

CiteSeerX

A new operation on sequences: The boustrophedon transform

N.J.A. Sloane

N.E. Young

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A New Operation on Sequences: The Boustrophedon Transform

AbstractA generalization of the Seidel–Entringer–Arnold method for calculating the alternating permutation numbers (or secant–tangent numbers) leads to a new operation on sequences, the boustrophedon transform

Millar, J.

Sloane, N.J.A.

Young, N.E.

Elsevier - Publisher Connector 

A New Operation on Sequences: The Boustrophedon Transform 

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Journal of Combinatorial Theory, Series A  TA2700
journal of combinatorial theory, Series A 76, 4454 (1996)
A New Operation on Sequences:
The Boustrophedon Transform
J. Millar,* N. J. A. Sloane, and N. E. Young*
Mathematical Sciences Research Center, ATHT Bell Laboratories,
Murray Hill, New Jersey 07974
Communicated by the Managing Editors
Received October 9, 1995
A generalization of the SeidelEntringerArnold method for calculating the alter-
nating permutation numbers (or secanttangent numbers) leads to a new operation
on sequences, the boustrophedon transform.  1996 Academic Press, Inc.
1. INTRODUCTION
Let En, k (nk0) denote the number of permutations of [1, 2, ..., n+1]
which alternately fall and rise (always starting with a fall), and start with
k+1. These numbers have a long history (see the references), but we follow
Poupard [Pou82] and call them the Entringer numbers. They satisfy the
recurrence [Ent66, first lemma]
E0, 0=1, En, 0=0 (n1), En+1, k+1=En+1, k+En, n&k (nk0).
(1.1)
If these numbers are displayed in a triangular array with rows written alter-
nately right to left and left to right, in boustrophedon (or ‘‘ox-plowing’’) manner:
E00 1
E10  E11 0  1
E22  E21  E20 = 1  1  0
E30  E31  E32  E33 0  1  2  2
E44  E43  E42  E41  E40 5  5  4  2  0
} } } } } }
(1.2)
article no. 0087
44
0097-316596 18.00
Copyright  1996 by Academic Press, Inc.
All rights of reproduction in any form reserved.
* Present address: Math. and Computer Science Department, Dartmouth College, Hanover,
New Hampshire.
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then the entries are filled in by the rule that each row (after the zeroth)
begins with a 0 and every subsequent entry is the sum of the previous entry
in the same row and the entry above it in the previous row.
The earliest reference we have seen for this elegant observation is Arnold
[Arn91], who refers to (1.2) as the EulerBernoulli triangle, but it may
well be of much older origin. Dumont [Dum95] refers to (1.2) as the
SeidelEntringerArnold triangle, referring to Seidel [Sei77].
The numbers En :=En, n appearing at the ends of the rows in (1.2) give
the total number of permutations of [1, 2, ..., n] that alternately fall and
rise, i.e. the number of ‘‘down-up permutations’’ of n things. The history of
these numbers goes back to Andre [And79], [And81], [Com74],
[Sch61]. They have exponential generating function (e.g.f.)
E(x)= :

n=0
En
xn
n!
=sec x+tan x. (1.3)
Conway and Guy [CG96] call (1.2) the zig-zag triangle and the En the
zig-zag permutation numbers. The Entringer numbers have also been
shown to enumerate several classes of rooted planar trees as well as other
mathematical objects [Arn91], [Arn92], [Kem33], [KPP94], [Pou82].
Guy [Guy95] observed that if the entries at the beginnings of the rows
in (1.2) are changed from 1, 0, 0, 0, ... to say 1, 1, 1, 1, 1, ..., or 1, 2, 4,
8, 16, ..., etc., then the numbers at the ends of the rows form interesting-
looking sequences not to be found in [SP95]. Using 1, 1, 1, ... for example
the triangle becomes
1
(1.4)
1 2
4 3 1
1 5 8 9
24 23 18 10 1
1 25 48 66 76 77
} } }
yielding the sequence
1, 2, 4, 9, 24, 77, 294, 1309, ... . (1.5)
Guy asked if anything could be said about generating functions or com-
binatorial interpretations for these sequences. The purpose of this note is to
answer this question.
45THE BOUSTROPHEDON TRANSFORM
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2. THE BOUSTROPHEDON TRANSFORM
Given a sequence1 a=(a1 , a1 , a2 , ...) we define its boustrophedon trans-
form to be the sequence b=(b0 , b1 , b2 , ...) produced by the triangle
a0=b0
a1  b1=a0+a1
b2=a1+a2+b1  a2+b1  a2 (2.1)
a3  a3+b2  a2+a3+b1+b2  b3=2a2+a3+b1+b2
} } }
when it is filled in using the rule described in Section 1. Formally, the
entries Tn, k (nk0) in the triangle are defined by
Tn, 0=an (n0),
(2.2)
Tn+1, k+1=Tn+1, k+Tn, n&k (nk0),
and then
bn=Tn, n (n0).
Although many operations on sequences have been studied in the past (see
[BS95] and the references therein), this transformation appears to have
been overlooked.
Theorem 1. The boustrophedon transform b of a sequence a is given by
bn= :
n
k=0 \
n
k+ ak En&k , (n0), (2.3)
an= :
n
k=0
(&1)n&k \nk+ bkEn&k , (n0), (2.4)
and the e.g.f.’s of b and a are related by
B(x)=(sec x+tan x) A(x). (2.5)
Proof. We redraw (2.1) as a directed graph 1 whose nodes are labeled
by the numbers Tn, k (see Fig. 1). Let ?(n, k, i ) denote the number of paths
46 MILLAR, SLOANE, AND YOUNG
1 In this paper we consider only integer-valued sequences, although the transformation can
be applied to sequences over any ring.
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(2.6)
Fig. 1. Directed graph 1 underlying the boustrophedon transform.
in 1 from the node labeled Ti, 0 to the node labeled Tn, k . It follows from
the rule for constructing the triangle that the numbers Tn, k are given by
Tn, k= :
n
i=0
?(n, k, i )ai . (2.7)
From Section 1 we know that the boustrophedon transform of the
sequence 1, 0, 0, 0, ... is E0 , E1 , E2 , E3 , ..., and so (from (2.7))
En=?(n, n, 0) (n0). (2.8)
We will give a direct proof of this (although of course it is known result,
cf. [Arn92]), in order to establish a bijection between paths in 1 and
up-down permutations.
Proposition 1. ?(n, n, 0) is equal to En , the number of down-up per-
mutations of [1, 2, ..., n].
Proof. Let P be a path in 1 from the top node to the node labeled Tn, n .
(Figure 2 shows an example for n=5.) Let Ti, f (i ) be the label of the node
where P arrives at level i (1f (i )in). We construct a box diagram to
represent P by the following procedure (see Fig. 3). The bottom row con-
tains n boxes labeled 1, ..., n from left to right (if n is even) or from right
to left (if n is odd). The box labeled f (n) is starred. We now repeatedly
place a row of boxes above the empty boxes, putting a star in the f (i )th
box, always counting from the left if i is even or from the right if i is odd,
for i=n&1, n&2, ..., 1.
47THE BOUSTROPHEDON TRANSFORM
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Fig. 2. A path from T0, 0 to T5, 5 .
We convert the box diagram into a permutation of [1, ..., n] by reading
the rows from the bottom up and recording the number at the foot of the
column containing the star. (The permutation corresponding to the above
example is (3, 1, 4, 2, 5).) We omit the easy verification that this process
defines a bijection between paths and down-up permutations. K
Proposition 2.
?(n, n, k)=\nk+ En&k , for 0kn.
Sketch of Proof. Consider a path from the node labeled Tk, 0 to the
node labeled Tn, n , such as the path from T4, 0 to T9, 9 shown in Fig. 4. The
procedure used in the proof of Proposition 1 converts this into a box
diagram, which for this example is shown in Fig. 5. The columns that do
not contain stars identify one of the ( nk) k-subsets of [1, ..., n], while the
starred columns themselves form a box diagram (in this case it is that
shown in Fig. 3) that identifies a down-up permutation of [1, ..., n&k]. K
Fig. 3. Box diagram corresponding to path P in Fig. 2.
48 MILLAR, SLOANE, AND YOUNG
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Fig. 4. A path from T4, 0 to T9, 9 .
From Proposition 2 and (2.7) we obtain
bn=Tn, n= :
n
k=0 \
n
k+ En&kak ,
which establishes (2.3). Eqs. (2.5) and (2.4) now follow immediately. This
completes the proof of the theorem. K
Remark. With only a little more effort we can determine all the
‘‘boustrophedon numbers’’ ?(n, k, i ). Note that ?(n, 0, i )=0 for n1,
0in&1, and ?(n, 0, n)=1.
Fig. 5. Box diagram for path shown in Fig. 4.
49THE BOUSTROPHEDON TRANSFORM
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Proposition 3. For n1, 0kn&1,
?(n, k, 0)=En, k= :
[(k&1)2]
r=0
(&1)r \ k2r+1+ En&2r&1.
Proof. ?(n, k, 0)=En, k follows from (2.7) and the definition of En, k (see
(1.2)), and the formula for En, k is given in [Ent66]. K
Remark. If the path is extended to reach the node labeled Tn+1, n+1 ,
the corresponding box diagram has the same format as those arising in
Proposition 1, except that the star in the last row is constrained to appear
in the box labeled k+1.
Proposition 4. For n2, 0<k<n, 0<in,
?(n, k, i )= :
min[k, n&i]
s=0 \
k
s+\
n&k
n&i&s+ ?(n&i, s, 0). (2.9)
Sketch of Proof. Consider a path P from Ti, 0 to Tn, k , and complete it
to a path Q from Ti, 0 to Tn+1, n+1 by extending P by a downward sloping
edge and a series of horizontal edges, as illustrated in Fig. 6. We form the
Fig. 6. Path from T2, 0 to T6, 4 (solid line) and its continuation to T7, 7 (broken line).
50 MILLAR, SLOANE, AND YOUNG
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Fig 7. (a) Box diagram corresponding to path shown in Fig. 6. (b) Reduced box
diagram, representing path of type ?(4, 3, 0).
box diagram for Q, as in Proposition 2 (see Fig. 7a). After deleting all the
unstarred columns we obtain the box diagram for a path of type
?(n&i, s, 0), for some s (Fig. 7b).
In the box diagram for Q itself, the star in the last row divides the
remaining stars into two sets of sizes s (to the right) and n&i&s (to
the left), and the binomial coefficients in (2.9) count the ways in which the
corresponding columns can be selected. K
Propositions 14 together express all the boustrophedon numbers in
terms of the En’s, and via (2.7) give an explicit formula for every entry
in the triangle (2.1).
3. COMBINATORIAL INTERPRETATIONS AND EXAMPLES
Equation (2.3) yields many possible combinatorial interpretations for the
numbers bn . For example, if an is the number of arrangements of n labeled
objects so that they have some property Q, then bn is the number of ways
of dividing n objects into two groups so that the first group has property
Q and the second forms a down-up sequence. Since En is also the number
of ordered binary trees on n nodes (cf. [Pou82], [KPP94]), other inter-
pretations for the bn can be given in terms of graphs.
Example 1. We can see now that (1.5) has e.g.f. ex(sec x+tan x), and
that the n th term of this sequence gives the number of ways we can form
a down-up sequence of some length l0 from [1, ..., n]. E.g. for n=3 there
are 9 possibilities: ,, 1, 2, 3, 21, 31, 32, 213, 312.
Example 2. The boustrophedon transform of the Bell numbers (cf.
[SP95], Fig. M4981) produces the sequence 1, 2, 5, 16, 60, 258, ..., whose
nth term gives the number of ways to take blocks labeled 1, ..., n and to
partition some of them into heaps and to arrange the rest so they form a
down-up sequence.
51THE BOUSTROPHEDON TRANSFORM
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Fig. 8. The double-ox transform of 1, 1, 1, ... is 1, 3, 9, 35, 177, ... .
Example 3. The boustrophedon transform of the En sequence shifted
one place to the left is the same sequence shifted two places to the left:
1
1  2
5  4  2
5  10  14  16
61  56  46  32  16
} } }
In view of Theorem 1, this means the e.g.f. E(x) satisfies
E(x) E$(x)=E"(x).
The initial conditions E0=E1=1 then give E(x)=sec x+tanx as the solution.
Example 4. The sequence 1, 0, 1, 1, 2, 6, 17, 62, 259, 1230, ... is the
lexicographically earliest sequence that begins with 1 and shifts two places
left under the boustrophedon transform. (Examples 3 and 4 are both eigen-
sequences for this transform, in the notation of [BS95].) We do not know
of any combinatorial interpretation for these numbers.
Example 5: The Double-Ox Transform. Generalizing some examples
of Arnold ([Arn92], see also [Dum95]), we consider two oxen plowing
separate fields with a messenger that takes the output at the end of one row
and rushes it to be used by the other ox as input to the next row. For
example, if the initial sequence (shown in italics in Fig. 8) is 1, 1, 1, ..., this
produces the output sequence (shown in bold) 1, 3, 9, 35, 177, 1123, ... .
Less colorfully, let a=a0 , a1 , ... be the initial sequence, m=m0 , m1 , ...
the middle (or messenger) sequence, and b=b0 , b1 , ... the transformed
sequence. We define two triangles of numbers [Ln, k] and [Rn, k], with
0kn, by
L2i, 0=a2i , R2i+1, 0=a2i+1 ,
L2i, 2i=R2i, 0=m2i , L2i+1, 0=R2i+1, 2i+1=m2i+1 ,
L2i+1, 2i+1=b2i+1 , R2i, 2i=b2i ,
52 MILLAR, SLOANE, AND YOUNG
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and
Ln+1, k+1=Ln+1, k+Ln, n&k , Rn+1, k+1=Rn+1, k+Rn, n&k .
We were happy to find that Theorem 1 leads to an equally simple
description of this transformation. The proof is left to the reader.
Theorem 2. The e.g.f. ’s of a, m and b are related by
M(x)=
1
cos x&sin x
A(x),
B(x)=
cos x+sin x
cos x&sin x
A(x).
ACKNOWLEDGMENTS
We thank Richard Guy for suggesting the problem, and J. H. Conway, I. M. Pak, and S.
Plouffe for helpful discussions. A referee suggested that since the boustrophedon and double-
ox transforms are associated with the root systems An and Cn , respectively (cf. [Arn92]), it
would be worthwhile studying analogues of these transforms for other root systems.
REFERENCES
[And79] D. Andre , De veloppements de sec x et de tang x, C. R. Acad. Sci. Paris 88 (1879),
965967.
[And81] D. Andre , Sur les permutations alterne es, J. Math. Pures Appl. 7 (1881), 167184.
[Arn91] V. I. Arnold, Bernoulli-Euler updown numbers associated with function singularities,
their combinatorics and arithmetics, Duke Math. J. 63 (1991), 537555.
[Arn92] V. I. Arnold, The calculus of snakes and the combinatorics of Bernoulli, Euler and
Springer numbers of Coxeter groups, Uspekhi Mat. Nauk. 47, No. 1 (1992), 345;
Russian Math. Surveys 47 (1992), 151. [Engl. transl.]
[BS95] M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear
Algebra Appl. 226228 (1995), 5772.
[Com74] L. Comtet, ‘‘Advanced Combinatorics,’’ Reidel, Dordrecht, 1974.
[CG96] J. H. Conway and R. K. Guy, ‘‘The Book of Numbers,’’ Springer-Verlag,
New York, in press.
[Dum95] D. Dumont, Further triangles of SeidelArnold type and continued fractions
related to Euler and Springer numbers, Adv. Appl. Math. 16 (1995), 275296.
[Ent66] R. C. Entringer, A combinatorial interpretation of the Euler and Bernoulli
numbers, Nieuw. Arch. Wisk. 14 (1966), 241246.
[Guy95] R. K. Guy, personal communication, 1995.
[Kem33] A. Kempner, On the shape of polynomial curves, To^hoku Math. J. 37 (1933),
347362.
53THE BOUSTROPHEDON TRANSFORM
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[KPP94] A. G. Kuznetzov, I. M. Pak, and A. E. Postnikov, Increasing trees and alternating
permutations, Uspekhi Mat. Nauk. 49, No. 6 (1994), 79110, [in Russian]; Russian
Math. Surveys 49, No. 6 (1994), 79114 [Engl. transl.].
[Pou82] C. Poupard, De nouvelles significations enumeratives des nombres d’Entringer,
Discrete Math. 38 (1982), 265271.
[Sch61] I. J. Schwatt, ‘‘An Introduction to the Operations with Series,’’ 2nd ed., esp.
Chap. XV, Chelsea, New York, 1961.
[Sei77] L. Seidel, U ber eine einfache Entstehungsweise der Bernoullischen Zahlen und
einiger verwandten Reihen, Sitzungsber. Mu nch. Akad. 4 (1877), 157187.
[SP95] N. J. A. Sloane and S. Plouffe, ‘‘The Encyclopedia of Integer Sequences,’’ Academic
Press, San Diego, 1995.
54 MILLAR, SLOANE, AND YOUNG


A generalization of the Seidel-Entringer-Arnold method for calculating the alternating permutation numbers (or secant-tangent numbers) leads to a new operation on sequences, the boustrophedon transform. © 1996 Academic Press, Inc. * Present address: Math. and Computer Science Department, Dartmouth College, Hanover, New Hampshire

Millar, J

Sloane, NJA

Young, NE

eScholarship - University of California

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.241.529