MATCHING PARTITION GENERATING FUNCTIONS
Match the description of the set of partitions with its generating function. Recall that a partition
of n is a sum λ1 + λ2 + · · · + λr = n. The order of the sum doesn’t matter so to avoid confusion we
assume that λ1 ≥ λ2 ≥ · · · ≥ λr . The λi are called the parts of the partition. r here is the number
of parts of the partition or the length of the partition. The sizes of the parts are the values λi . The
size of the partition is the sum of the sizes of all the parts (in this case n). Parts are called distinct
if they are not equal to each other. The number of parts of a given size refers to the number of
times that a value appears as a part.
Note: There 17 generating functions and 18 descriptions listed below because two of the descriptions
have the same generating function.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
the number of partitions of n
the number of partitions of n into exactly k parts
the number of partitions of n with parts of size k only
the number of partitions of n with parts of size less than or equal to k
the number of partitions of n with distinct parts
the number of partitions of n with odd parts
the number of partitions of n with distinct odd parts
the number of partitions of n with even parts
the number of partitions of n with distinct even parts
the number of partitions of n into parts congruent to 1 or 4 modulo 5
the number of partitions of n with at most 4 parts of any given size
the number of partitions of n with (for each i) the number of size i is less than i.
the number of partitions of n and for each i, if there is a part of size i then it occurs an odd
number of times.
the number of partitions of n and for each i, the parts of size i occur an even number of times.
the number of partitions of n with only odd parts and the number of parts of any given size is
even.
the number of partitions of n with odd parts and at most 4 parts of any given size
the number of partitions of n with even parts and at most 4 parts of any given size
the number of partitions of n with at least one even part
(a)
Y
i≥1
1
1 − xi
(b)
Y
(1 + x2i−1 )
i≥1
(c)
1
Y
i≥0
(1 −
x5i+1 )(1
1
− x5i+4 )
2
MATCHING PARTITION GENERATING FUNCTIONS
(d)
Y 1 − x i2
i≥1
1 − xi
(e)
k
Y
i=1
1
1 − xi
(f)
xk
k
Y
i=1
1
1 − xi
(g)
Y
(1 + x2i )
i≥1
(h)
Y 1 − x10i−5
i≥1
1 − x2i−1
(i)
x2 Y 1
1 − x2
1 − xi
i≥1
(j)
Y
i≥1
1
1 − x4i−2
(k)
Y
i≥1
1
1 − x2i
(l)
Y 1 − x5i
i≥1
1 − xi
(m)
Y
1+
i≥1
xi
1 − x2i
(n)
1
1 − xk
(o)
Y
i≥1
1
1 − x2i−1
(p)
Y
(1 + xi )
i≥1
(q)
Y 1 − x10i
i≥1
1 − x2i