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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:

Restricted Partitions:

Odd partitions of $8$:

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

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

$$\displaystyle \sum_{n=0}^\infty p(n) x^n = \prod_{k=1}^\infty \frac{1}{1-x^k}$$ 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

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

$$p_n^s=p_n^t$$

$$p_n^o=p_n^d$$

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

ferrers.png

8. Statistics

We have the following 131 statistics on Integer partitions 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 largest 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 distinct parts of the 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 when ....
St000179 Integer partitions ⟶ ℤ
The product of the hook lengths of the integer partition.
St000182 Integer partitions ⟶ ℤ
The number of permutations whose cycle type is the given integer partition.
St000183 Integer partitions ⟶ ℤ
The side length 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
The number of standard Young tableaux for an integer partition such that no two c....
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 largest part of the 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.
St000993 Integer partitions ⟶ ℤ
The multiplicity of the largest part of an integer partition.
St000995 Integer partitions ⟶ ℤ
The largest even part of an integer partition.
St000997 Integer partitions ⟶ ℤ
The even-odd crank of an integer partition.
St001055 Integer partitions ⟶ ℤ
The Grundy value for the game of removing cells of a row in an integer partition.....

9. Maps

We have the following 16 maps from and to Integer partitions 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
reading tableau
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:

For many combinatorial statistics and bijections see [Pak02]

10. References

11. Sage examples