A combinatorial proof of Schur's 1926 partition theorem

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A combinatorial proof of Schur's 1926 partition theorem

Citation for published version (APA):

Post, K. A. (1978). A combinatorial proof of Schur's 1926 partition theorem. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 7815). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1978

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-- - i

EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics

Memorandum 1978-15

December 1978

A combinatorial proof of Schur's 1926 partition theorem

University of Technology Department of Mathematics PO Box 513, Eindhoven The Netherlands. by K.A. Post

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by

K.A. Post

Schur's theorem

Given positive integers rand m such that r <

~2

' let C (n) denote the

num-r,m

ber of partition of n into distinct parts congruent to ±r (mod m) and let

D (n) denote the number of partitions of n into distinct parts congruent

r,m .

to 0, ±r (mod m) with minimal difference m, and minimal difference 2m between

multiples of m. Then C (n)

r,m D r,m (n) for all n.

Let m and r be positive integers, m > 2r. Let

be a partition of the positive integer n into positive parts, that are con-gruent to ±r (mod m) .

We subdivide the sequence (a.) from left to right into blocks of size 2

(pre-1.

ferably) and 1 such that no two elements with difference ~ m ever belong to

the same block. This subdivision is obviously unique.

Example. m

=

5, r

=

1. The sequence

(a.)

=

(4,11,14,16,21,26,29,36,39,41)

1.

is a partition of n

=

237 and is subdivided into

4111,14116121126,29136,39141

For all j let b . denote the sum of the elements in block j, and let c. be

de-J J

fined as

C. := b. - ( j - l ) m .

J J

In our example we have therefore (c

j ) = (4,20,6,6,35,50,11)

The sequence (c.), obtained in this way has the following properties: J

Property 1. For all j we have c. _ 0, r or -r (mod m).

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2

-Property 2. For all j we have:

i) c,

_-

±r (mod m) ~ (c, _{originates from a block of size}1 containing the

J J

element c, + (j - 1)m. J

ii) c , - 0 (mod m) ~ _{(c, originates from a block of size}2 containing the

J J _c _, _{+ (j-l)m} _c _, _{+ (j - l)m} elements

l

J J and

r

J

1 .

2 r 2 r

In this assertion we use the notation

l

g J and

r

g

1

to denote Max{g E M

I

x < g}

r r

and Min{x E M

I

x > g} respectively, where M := {x E ~

I

x

=

±r (mod m)}.

Property 3. For all j we have

i) c, _ 0 (mod m) } J ~ c _ 0 (mod m) j+l c_{j +}₁ ii) c , -J c j +1 ±r (mod m) } .. c , 1 J+

-

±r (mod m) iii) c , -J c j+1 0 (mod m) m)} .. c ,

l

J - ±r (mod ~ c, + m J ~ c, J + (j - 1) m 2 J r + m :'0: cJ+ , 1 + jm iv) c = ±r (mod m) }

lC

j + 12+ jm_Jr j - ~ c,

+

jm :'0: c j+1

=

0 (mod m) J Property 4.

i) The subsequence of those c , which are _ ±r (mod m) is a non-decreasing

J

sequence.

ii) The subsequence of those c , that are _ 0 (mod m) is increasing (with dif-J

ferences ~ m) .

Property 5. For all j' > j we have i) ii) c_t - 0 (mod m) (j ~ t < jl)} .. C ' I

=

±r (mod m) J c,

-l

J c t

=

±r (mod m) (j :'0: t :'0: ~ c, :'0: C ' I

=

0 (mod m) J J (j - l)m

2 J

r :'0: C_J, I C ' I - (j I - 1) m

l

~---:2---r J

J

The proof of these properties is straightforward. Moreover, any sequence (c,)

J

of positive integers, having property 1 and 3 originates from a unique se-quence (a,) by the construction given above.

1

Now let (d,) be the non-decreasing rearrangement of (c,), and for all j let

J J

finally e , be given as

J

e, := d,

+

(j-l)m

J J

Then (e,) has the following property (*).

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-(e .) is an increasing sequence of positive numbers congruent to 0, +r J

or -r (mod m) with differences ~ m, and differences between multiples of m being at least 2m. Moreover, (e,) is a partition of n.

J

In our example we obtain

(e,) = (4,11,16,26,40,60,80)

J

Now we shall show that the construction of (e,) from (a,) is reversible, i.e.

J J

given any sequence (e,) satisfying (*) there exists a (unique)

J part~ tion (a~ ,)

of n into distinct positive parts congruent to ±r(mod m) that yields (e .) by J the construction. All steps are immediate except how to ,obtain the sequence

(c ,) from the sequence (d,) by interlacing the subsequences of terms

=

0 (mod m)

J J

and of terms

=

±r (mod m) in the latter.

-Let d

1 < d2 < ••. be the subsequence of those dj which are congruent to 0 (mod m), and let d

1 ~ d2 ~ •.. be the subsequence of those d, that are congru-J _ ~

ent to ±r (mod m). Now property 5 will be the guide to interlace (~) and (~).

For c

1 there are two candidates, d1 and d1. According to property 5 we must decide in favour of -d

1, ;f ld21Jr ~ < - d l ' an d ~n ' f avour 0 f

~d

1 ~n ' t h e case were h - pm

2 J

r for some positive integer p (hence if we observe that

-d 1 d 1 ~ l

-- m

----2--J r). These two criteria turn out to be exactly complementary. d 1 d 1 ~ l

-

-Hence, c

1 is uniquely determined. Now we proceed by induction: Let d1, ... ,ds_1

~

and d₁, ... ,d

t_1 be chosen in the sequence (cj ) to_form t~e elements

c

1,c2' · · · , cj_1• For cj there are two candidates, ds and dt . According to pro-d - (j-1)m

perty 5 we must decide in favour of d , if l--s----~---J <

d

and in favour

s

2

r - t

-

-d - p m d - jm

O f dt ; f d

t l s J < l s J )

~ ~ 2 r for some p ~ j (hence if we see that d

t 2 r ·

Again, these conditions are exactly complementary, so that c, is uniquely de-J

termined. The properties 3(i+iv) are now also valid for (c,) so that the basic J

construction is uniquely inverted.

References

[1J G.H. Hardy and E.M. Wright, An introduction to the theory of numbers, 4th ed. Oxford (1960).

[2J I.J. Schur, "Zur additiven Zahlentheorie". Gesammelte Abhandlungen, vol. 3, p. 43-50, Springer, Berlin (1973).