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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 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
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