Combinatorics. Problem Set 6. Partitions
Seminar problems
A decomposition of n (with k parts) is a representation of n as n = a1 + a2 + · · · + ak , where all ai are
nonnegative integers. The order here is important: two decompositions n = a1 + · · · + ak = b1 + · · · + bk
are said to be distinct iff ai 6= bi for at least one index i.
Problem 6.1. Compute the number Decompn,k of decompositions of n with k parts.
Problem 6.2. What is the generating function of decompositions,
∞
X
X
Decompn,k · xn =
n=0
xn
?
decompositions n = a1 + · · · + ak
Here k ≥ 1 is fixed.
A composition of n is a representation n = r1 +· · ·+r` , where ri are positive integers and ` is arbitrary. The
order here is important. For example, there are four decompositions of 3: 3 = 3 = 2 + 1 = 1 + 2 = 1 + 1 + 1.
n−1
Problem 6.3. Show that the number of compositions of n with exactly ` parts is
. (Hint: interpret
`−1
this as some number of decompositions).
A partition of n is a representation of the form n = λ1 + · · · + λ` , where λi are positive integers,
and the order is irrelevant. Thus, we may assume λ1 ≥ · · · ≥ λ` . For example, the partitions of 5 are
5 = 5 = 4 + 1 = 3 + 2 = 3 + 1 + 1 = 2 + 2 + 1 = 2 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1. Partitions are represented
by Young diagrams. For example, the partition λ = (6, 3, 3) corresponds to the diagram
Problem 6.4. Compute the number
of n into parts equal to 1. Let this be pn (1). What is
P∞ of partitions
n
the generating function P (1; q) = n=0 pn (1)q ?
Problem 6.5. Compute the number
of n into parts equal to 1. Let this be pn (2). What is
P of partitions
n
p
(2)q
?
the generating function P (2; q) = ∞
n=0 n
Problem 6.6. Show that the generating function of the number of partitions of n into parts which are
equal to 1 or 2, is P (1; q)P (2; q).
Problem 6.7. Prove Euler’s theorem: the generating function of the number pn of all partitions of n is
∞
X
X
1
P (q) :=
pn q n =
q number of boxes of λ =
.
2
3 )(1 − q 4 ) . . .
(1
−
q)(1
−
q
)(1
−
q
n=0
all partitions λ
Problem 6.8. Give a combinatorial interpretation to the Euler’s theorem written in the following form:
∞
P
pn q n = (1 + q + q 2 + . . . )(1 + q 2 + q 4 + . . . )(1 + q 3 + q 6 + . . . ) . . . .
n=0
Problem 6.9. Show that lim Cn (q) = P (q), where Cn (q) are the q-Catalan numbers.
n→+∞
Homework
Problem 6.10. (2 ) Show that the generating function in two variables of decompositions has the form
∞ P
∞
P
q
Decompn,k · xn q k =
.
1−q−x
n=0 k=1
1
2
Combinatorics. Problem Set 6. Partitions
Problem 6.11. (1 ) Show that the number of decompositions of fixed n (which is
infinite.
P∞
k=1
Decompn,k ) is
Problem 6.12. (1 ) List all the compositions of 4 and 5. What is the number of such compositions?
Problem 6.13. (3 ) Compute the number of all possible compositions of n.
Problem 6.14. (4 ) What is the number of ways to change a dollar using 1, 5, 10, and 25-cent coins (e.g.,
1 dollar = 25 + 25 + 10 + 10 + 10 + 10 + 5 + 5; the order of coins does not matter) ?
Problem 6.15. (3 ) Compute the generating function for the number of partitions of n into ≤ 4 parts.
(Hint: use Problems 6.4–6.7 and transposition of Young diagrams.1)
Problem 6.16. (2 ) Compute the number of partitions of 5, 6, 7, and 8.
Problem 6.17. (3 ) Arguing as in Problem 6.8, show that the generating function of the number of
partitions of n into distinct parts is
X
q number of boxes in λ = (1 + q)(1 + q 2 )(1 + q 3 )(1 + q 4 ) . . . .
all partitions λ with distinct parts
Problem 6.18. (4 ) Using the fact that any number has a unique representation in decimal arithmetic
1
system, show that (1 + q + q 2 + · · · + q 9 )(1 + q 10 + q 20 + · · · + q 90 ) . . . =
.
1−q
Problem 6.19. (7 ) Arguing as in problem 6.18 for the binary arithmetic system, show that
1
.
(1 + q)(1 + q 2 )(1 + q 3 )(1 + q 4 ) · · · =
3
(1 − q)(1 − q )(1 − q 5 )(1 − q 7 ) . . .
Problem 6.20. (4 ) From the previous problem deduce that the number of partitions of n into distinct
parts is the same as the number of partitions of n into odd (not necessary distinct) parts.
Problem 6.21. (3 ) Show that the number of partitions of n is equal to the number of partitions of 2n
with exactly n (positive) summands.
Supplementary problems
n
Problem 6.22. (7 ) Let k q be the q-binomial coefficient. That is,
hni
X
(1.1)
=
q the area below the path w ,
k q
up-right lattice paths w from (0, 0) to (k, n − k)
[n]!q
, where [a]!q := 1 · (1 + q)(1 + q + q 2 )(1 + q + q 2 + q 3 ) . . . (1 + q + q 2 + · · · + q a−1 )
[k]!q [n − k]!q is the q-factorial. Note that nk q is a polynomial for any n, k.
Show that for any fixed m, the coefficient by q m in the polynomial nk q stabilizes as n, k → +∞. That
is, for all n ≥ n0 and k ≥ k0 this coefficient is the same. (Hint: use the combinatorial definition (1.1).)
(4 ) Give a combinatorial interpretation of this stabilized coefficient.
and also
n
k q
=
Problem 6.23. (10 ) q-Catalan numbers are defined as
X
Cn (q) :=
q the area between w and the path (1, 1, . . . , 1, −1, −1, . . . , −1) .
Dyck paths w of length 2n
Deduce a recurrence
Pn−1 formula for these q-Catalan numbers which for q = 1 becomes the known recurrence
formula Cn = k=0 Ck Cn−1−k .
1A
transposed Young diagram is obtained from the original diagram by reflection with respect to the diagonal.