# 1. Definition

An Integer Partition of $n \in \mathbf{N}$ is a unique way of writing n as the sum of integers. Partitions that differ in the order only are considered to be the same partition. Restricted partitions is a partition that is restricted under specified conditions.

# 2. Notation

By $\mathbf{P}_n$, we denote the collection of all integer partitions of $n$. We usually write $\lambda \in \mathbf{P}_n$ as $\lambda = (\lambda_1,\lambda_2, \ldots)$.

# 3. Examples

Partitions:

• $\mathbf{P}_1 = \{ [1]\}$

• $\mathbf{P}_2 = \{ [2],[1,1]\}$

• $\mathbf{P}_3 = \{ [3],[2,1],[1,1,1]\}$

• $\mathbf{P}_4 = \{ [4],[3,1],[2,2], [2,1,1], [1,1,1,1]\}$

Restricted Partitions:

Odd partitions of $8$:

• $7+1$

• $5+3$

• $5+1+1+1$

• $3+3+1+1$

• $3+1+1+1+1+1$

• $1+1+1+1+1+1+1+1$

Partitions of $8$ with distinct parts (Non-Repeating Integers):

• $8$

• $7+1$

• $6+2$

• $5+3$

• $5+2+1$

• $4+3+1$

# 4. Partition Function

The Partition Function $P(n)$ represents the number of partitions of the integer n. The first few values of $P(n)$ are $1,1,2,3,5,7,11,15,22,30,42,$ and so on (A000041 in OEIS)

$Q(n)$ is denoted as the number of partitions of distinct parts and of odd partitions. The first few values of $Q(n)$ are $1,1,1,2,3,4,5,6,7,8,10$ (A000009 in OEIS)

# 5. Generating Function

• $|\mathbf{P}_n| = p(n)$ counts the number of ways n can be partitioned. Hans Rademacher provided a way to determine $p(n)$ by introducing a series called Rademacher's series:

$$\displaystyle \sum_{n=0}^\infty p(n) x^n = \prod_{k=1}^\infty \frac{1}{1-x^k}$$ which can be expanded to $$(1 + x + x^2 + x^3 + ....)(1 + x^2 + x^4 + x^6 + ....)....$$ This is also known as the Euler's generating function which allows $p(n)$ to be calculated for any $n$. [Wil00]

# 6. Properties

• Let $p_n(r)$ be the number of partitions of $n$ in which the largest part is $r$. Let $q_n(r)$ be the number of partitions of $n-r$ which no part is greater than r. Then

$$p_n(r)=q_n(r)$$

• Let $p_n^s$ denote the number of self conjugate partitions of n and let $p_n^t$ be the number of partitions of n into distinct odd parts. Then

$$p_n^s=p_n^t$$

• Let $p_n^o$ be the number of partitions of n into odd parts, and let $p_n^d$ be the number of partitions of n into distinct parts. Then

$$p_n^o=p_n^d$$

• [Bru10]
• The elements of $\mathbf{P}_n$ index the number of irreducible representations of the symmetric group $\mathbf{S}_n$ over a field of characteristic $0$.

• The number of non-isomorphic abelian groups of order $n=\sum_ip_i^{j_i}$ where $p_i$ are distinct primes and $j_i$ are positive integers is given by $\prod_ip(j_i)$ A000688

• Congruence partitions are partitions that are similar under modulation. For example: $$p(5k+4)= 0 (mod\text{ }5)$$ $$p(7k+5)= 0 (mod\text{ }7)$$ $$p(11k+6)=0 (mod\text{ }11)$$ are just a couple examples proven by Srinivasa Ramanujan.

• The approximation formula for p(n): $$\displaystyle p(n) \sim \frac {1} {4n\sqrt3} \exp\left({\pi \sqrt {\frac{2n}{3}}}\right) \mbox { as } n\rightarrow \infty$$

# 7. Ferrer and Young diagrams

A very useful way to represent partitions is by the means of Ferrers diagram. Ferrers diagram, named after Norman Macleod Ferrers, is constructed by stacking left-justified rows of cells, where the number of cells in each row corresponds to the size of a part. The first row corresponds to the largest part, the second row corresponds to the second largest part and so on.

For example, partition $\lambda = (5,4,2,1)$ can be shown as

• The Young diagram is similar to the Ferrer diagram. Instead of dots, boxes are used. instead of ordering by rows, it is ordered by columns. The first columns corresponds to the largest part, the second columns corresponds to the second largest part and so on.

In this example, the Young diagram is $\lambda = (4,3,2,2,1)$

• A bijection can be created between either diagram into one with a unique diagram with distinct parts. Taking the above example we take the outside L dots/boxes and create a new row. The same thing for the second row and so on. The new diagram will have a Young diagram where $\lambda = (9,4)$

# 8. Statistics

We have the following 127 statistics in the database:

St000003
Integer partitions ⟶ ℤ
The number of standard Young tableaux of the partition.
St000010
Integer partitions ⟶ ℤ
The length of the partition.
St000046
Integer partitions ⟶ ℤ
The largest eigenvalue of the random to random operator acting on the simple modu....
St000048
Integer partitions ⟶ ℤ
The multinomial of the parts of a partition.
St000049
Integer partitions ⟶ ℤ
The number of set partitions whose sorted block sizes correspond to the partition....
St000063
Integer partitions ⟶ ℤ
The number of linear extensions of a certain poset defined from a partition \la....
St000088
Integer partitions ⟶ ℤ
The row sums of the character table of the symmetric group.
St000108
Integer partitions ⟶ ℤ
The number of partitions contained in the given partition.
St000137
Integer partitions ⟶ ℤ
The Grundy value of an integer partition.
St000142
Integer partitions ⟶ ℤ
The number of even parts of a partition.
St000143
Integer partitions ⟶ ℤ
The maximal repeated part of a partition.
St000145
Integer partitions ⟶ ℤ
The Dyson rank of a partition.
St000146
Integer partitions ⟶ ℤ
The Andrews-Garvan crank of a partition.
St000147
Integer partitions ⟶ ℤ
The largest part of an integer partition.
St000148
Integer partitions ⟶ ℤ
The number of odd parts of a partition.
St000149
Integer partitions ⟶ ℤ
The number of cells of the partition whose leg is zero and arm is odd.
St000150
Integer partitions ⟶ ℤ
The floored half-sum of the multiplicities of a partition.
St000159
Integer partitions ⟶ ℤ
The number of different parts of an integer partition.
St000160
Integer partitions ⟶ ℤ
Multiplicity of the smallest part of $\lambda$.
St000175
Integer partitions ⟶ ℤ
Degree of the polynomial counting the number of semistandard Young-tableaux of sh....
St000179
Integer partitions ⟶ ℤ
Product of hook lengths.
St000182
Integer partitions ⟶ ℤ
The number of permutations whose cycle type is the given integer partition.
St000183
Integer partitions ⟶ ℤ
The size of the Durfee square of an integer partition.
St000184
Integer partitions ⟶ ℤ
The size of the centralizer of any permutation of given cycle type.
St000185
Integer partitions ⟶ ℤ
The weighted size of a partition.
St000205
Integer partitions ⟶ ℤ
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and part....
St000206
Integer partitions ⟶ ℤ
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and inte....
St000207
Integer partitions ⟶ ℤ
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer ....
St000208
Integer partitions ⟶ ℤ
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer ....
St000212
Integer partitions ⟶ ℤ
Number of standard Young tableaux for an integer partition such that no two conse....
St000225
Integer partitions ⟶ ℤ
Difference between largest and smallest parts in a partition.
St000228
Integer partitions ⟶ ℤ
The size of a partition.
St000256
Integer partitions ⟶ ℤ
The number of parts from which one can substract 2 and still get an integer parti....
St000257
Integer partitions ⟶ ℤ
The number of distinct parts of a partition that occur at least twice.
St000275
Integer partitions ⟶ ℤ
Number of permutations whose sorted list of non zero multiplicities of the Lehmer....
St000278
Integer partitions ⟶ ℤ
The size of the preimage of the map 'to partition' from Integer compositions to I....
St000284
Integer partitions ⟶ ℤ
The Plancherel distribution on integer partitions.
St000318
Integer partitions ⟶ ℤ
The number of addable cells of the Ferrers diagram of an integer partition.
St000319
Integer partitions ⟶ ℤ
The spin of an integer partition.
St000320
Integer partitions ⟶ ℤ
The dinv adjustment of an integer partition.
St000321
Integer partitions ⟶ ℤ
The number of integer partitions of n that are dominated by an integer partition.....
St000345
Integer partitions ⟶ ℤ
The number of refinements of a partition.
St000346
Integer partitions ⟶ ℤ
The number of coarsenings of a partition.
St000377
Integer partitions ⟶ ℤ
The dinv defect of an integer partition.
St000378
Integer partitions ⟶ ℤ
The diagonal inversion number of an integer partition.
St000380
Integer partitions ⟶ ℤ
Half the perimeter of the largest rectangle that fits inside the diagram of an in....
St000384
Integer partitions ⟶ ℤ
The maximal part of the shifted composition of an integer partition.
St000459
Integer partitions ⟶ ℤ
The hook length of the base cell of a partition.
St000460
Integer partitions ⟶ ℤ
The hook length of the last cell along the main diagonal of an integer partition.....
St000473
Integer partitions ⟶ ℤ
The number of parts of a partition that are strictly bigger than the number of on....
St000474
Integer partitions ⟶ ℤ
Dyson's crank of a partition.
St000475
Integer partitions ⟶ ℤ
The number of parts equal to 1 in a partition.
St000477
Integer partitions ⟶ ℤ
The weight of a partition according to Alladi.
St000478
Integer partitions ⟶ ℤ
Another weight of a partition according to Alladi.
St000480
Integer partitions ⟶ ℤ
The number of lower covers of a partition in dominance order.
St000481
Integer partitions ⟶ ℤ
The number of upper covers of a partition in dominance order.
St000506
Integer partitions ⟶ ℤ
The number of standard desarrangement tableaux of shape equal to the given partit....
St000509
Integer partitions ⟶ ℤ
The diagonal index (content) of a partition.
St000510
Integer partitions ⟶ ℤ
The number of invariant oriented cycles when acting with a permutation of given c....
St000511
Integer partitions ⟶ ℤ
The number of invariant subsets when acting with a permutation of given cycle typ....
St000512
Integer partitions ⟶ ℤ
The number of invariant subsets of size 3 when acting with a permutation of given....
St000513
Integer partitions ⟶ ℤ
The number of invariant subsets of size 2 when acting with a permutation of given....
St000514
Integer partitions ⟶ ℤ
The number of invariant simple graphs when acting with a permutation of given cyc....
St000515
Integer partitions ⟶ ℤ
The number of invariant set partitions when acting with a permutation of given cy....
St000517
Integer partitions ⟶ ℤ
The Kreweras number of an integer partition.
St000531
Integer partitions ⟶ ℤ
The number of ways to place as many non-attacking rooks as possible on a Ferrers ....
St000532
Integer partitions ⟶ ℤ
The total number of rook placements on a Ferrers board.
St000533
Integer partitions ⟶ ℤ
The maximal number of non-attacking rooks on a Ferrers shape.
St000547
Integer partitions ⟶ ℤ
The number of even non-empty partial sums of an integer partition.
St000548
Integer partitions ⟶ ℤ
The number of different non-empty partial sums of an integer partition.
St000549
Integer partitions ⟶ ℤ
The number of odd partial sums of an integer partition.
St000566
Integer partitions ⟶ ℤ
The number of ways to select a row of a Ferrers shape and two cells in this row.
St000567
Integer partitions ⟶ ℤ
The sum of the products of all pairs of parts.
St000618
Integer partitions ⟶ ℤ
The number of self-evacuating tableaux of given shape.
St000620
Integer partitions ⟶ ℤ
The number of standard tableaux of shape equal to the given partition such that t....
St000621
Integer partitions ⟶ ℤ
The number of standard tableaux of shape equal to the given partition such that t....
St000644
Integer partitions ⟶ ℤ
The number of graphs with given frequency partition.
St000667
Integer partitions ⟶ ℤ
The greatest common divisor of the parts of the partition.
St000668
Integer partitions ⟶ ℤ
The least common multiple of the parts of the partition.
St000681
Integer partitions ⟶ ℤ
The Grundy value of Chomp on Ferrers diagrams.
St000697
Integer partitions ⟶ ℤ
The number of 3-rim hooks removed from an integer partition to obtain its associa....
St000698
Integer partitions ⟶ ℤ
The number of 2-rim hooks removed from an integer partition to obtain its associa....
St000704
Integer partitions ⟶ ℤ
The number of semistandard tableaux on a given integer partition with minimal max....
St000705
Integer partitions ⟶ ℤ
The number of semistandard tableaux on a given integer partition of n with maxima....
St000706
Integer partitions ⟶ ℤ
The product of the factorials of the multiplicities of an integer partition.
St000707
Integer partitions ⟶ ℤ
The product of the factorials of the parts.
St000708
Integer partitions ⟶ ℤ
The product of the parts of an integer partition.
St000712
Integer partitions ⟶ ℤ
The number of semistandard Young tableau of given shape, with entries at most 4.
St000713
Integer partitions ⟶ ℤ
The dimension of the irreducible representation of Sp(4) labelled by an integer p....
St000714
Integer partitions ⟶ ℤ
The number of semistandard Young tableau of given shape, with entries at most 2.
St000715
Integer partitions ⟶ ℤ
The number of semistandard Young tableau of given shape, with entries at most 3.
St000716
Integer partitions ⟶ ℤ
The dimension of the irreducible representation of Sp(6) labelled by an integer p....
St000749
Integer partitions ⟶ ℤ
The smallest integer d such that the restriction of the representation correspond....
St000752
Integer partitions ⟶ ℤ
The Grundy value for the game 'Couples are forever' on an integer partition.
St000755
Integer partitions ⟶ ℤ
The number of real roots of the characteristic polynomial of a linear recurrence ....
St000759
Integer partitions ⟶ ℤ
The smallest missing part in an integer partition.
St000770
Integer partitions ⟶ ℤ
The major index of an integer partition when read from bottom to top.
St000781
Integer partitions ⟶ ℤ
The number of proper colouring schemes of a Ferrers diagram.
St000783
Integer partitions ⟶ ℤ
The maximal number of occurrences of a colour in a proper colouring of a Ferrers ....
St000784
Integer partitions ⟶ ℤ
The maximum of the length and the number of parts of an integer partition.
St000810
Integer partitions ⟶ ℤ
The sum of the entries in the column specified by the partition of the change of ....
St000811
Integer partitions ⟶ ℤ
The sum of the entries in the column specified by the partition of the change of ....
St000812
Integer partitions ⟶ ℤ
The sum of the entries in the column specified by the partition of the change of ....
St000813
Integer partitions ⟶ ℤ
The number of zero-one matrices with weakly decreasing column sums and row sums g....
St000814
Integer partitions ⟶ ℤ
The sum of the entries in the column specified by the partition of the change of ....
St000815
Integer partitions ⟶ ℤ
The number of semistandard Young tableaux of partition weight of given shape.
St000835
Integer partitions ⟶ ℤ
The minimal difference in size when partitioning the integer partition into two s....
St000867
Integer partitions ⟶ ℤ
The sum of the hook lengths in the first column of an integer partition.
St000869
Integer partitions ⟶ ℤ
The sum of the hook lengths of an integer partition.
St000870
Integer partitions ⟶ ℤ
The product of the hook lengths of the diagonal cells in an integer partition.
St000897
Integer partitions ⟶ ℤ
The number of different multiplicities of parts of an integer partition.
St000901
Integer partitions ⟶ ℤ
The cube of the number of standard Young tableaux with shape given by the partiti....
St000913
Integer partitions ⟶ ℤ
The number of ways to refine the partition into singletons.
St000927
Integer partitions ⟶ ℤ
The alternating sum of the coefficients of the character polynomial of an integer....
St000928
Integer partitions ⟶ ℤ
The sum of the coefficients of the character polynomial of an integer partition.
St000929
Integer partitions ⟶ ℤ
The constant term of the character polynomial of an integer partition.
St000933
Integer partitions ⟶ ℤ
The number of multipartitions of sizes given by an integer partition.
St000934
Integer partitions ⟶ ℤ
The 2-degree of an integer partition.
St000935
Integer partitions ⟶ ℤ
The number of ordered refinements of an integer partition.
St000936
Integer partitions ⟶ ℤ
The number of even values of the symmetric group character corresponding to the p....
St000937
Integer partitions ⟶ ℤ
The number of positive values of the symmetric group character corresponding to t....
St000938
Integer partitions ⟶ ℤ
The number of zeros of the symmetric group character corresponding to the partiti....
St000939
Integer partitions ⟶ ℤ
The number of characters of the symmetric group whose value on the partition is p....
St000940
Integer partitions ⟶ ℤ
The number of characters of the symmetric group whose value on the partition is z....
St000941
Integer partitions ⟶ ℤ
The number of characters of the symmetric group whose value on the partition is e....
St000944
Integer partitions ⟶ ℤ
The 3-degree of an integer partition.
St000992
Integer partitions ⟶ ℤ
The alternating sum of the parts of an integer partition.

# 9. Maps

We have the following 16 maps in the database:

Mp00021
Cores ⟶ Integer partitions
to bounded partition
Mp00022
Cores ⟶ Integer partitions
to partition
Mp00027
Dyck paths ⟶ Integer partitions
to partition
Mp00037
Graphs ⟶ Integer partitions
to partition of connected components
Mp00040
Integer compositions ⟶ Integer partitions
to partition
Mp00042
Integer partitions ⟶ Standard tableaux
initial tableau
Mp00043
Integer partitions ⟶ Dyck paths
to Dyck path
Mp00044
Integer partitions ⟶ Integer partitions
conjugate
Mp00045
Integer partitions ⟶ Standard tableaux
Mp00060
Permutations ⟶ Integer partitions
Robinson-Schensted tableau shape
Mp00077
Semistandard tableaux ⟶ Integer partitions
shape
Mp00079
Set partitions ⟶ Integer partitions
shape
Mp00083
Standard tableaux ⟶ Integer partitions
shape
Mp00095
Integer partitions ⟶ Binary words
to binary word
Mp00108
Permutations ⟶ Integer partitions
cycle type
Mp00110
Posets ⟶ Integer partitions
Greene-Kleitman invariant

Maps not yet considered are:

• This corresponds to the unique Dyck path whose complement is the partition for which the partition touches the diagonal. If $w$ is a $n$-Dyck word, then the Dyck path corresponding to $w$ can be regarded as a lattice path in the northeastern half of an $n \times n$-square. The region to the northeast of this Dyck path can be regarded as a partition. It is called the partition corresponding to the Dyck word $w$. For every partition $\lambda$ and every non-negative integer $n$, there exists at most one $n$-Dyck word $w$ such that the partition corresponding to $w$ is $\lambda$. This method computes this $w$ for a given $\lambda$ and $n$. If $n$ is not specified, this method computes the $w$ for the smallest possible $n$ for which such an $w$ exists. For example minimal dyck word of partition $\lambda = (5,4,2,1)$ is $(1,1,0,1,0,1,0,0,1,0,1,0)$

For many combinatorial statistics and bijections see [Pak02]

# 10. References

• [Bru10] R. Brualdi, Partition numbers, Combinatorics Fifth Edition (2010).
• [Pak02] I. Pak, Partition bijections, a survey, Ramanujan J. 12(1) (2006) p. 5-75.
• [Wil00] S. Wilf, Basic generating functions, Lectures on Integer Partitions, (2000).