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65-1850: Schur polynomials are indexed by integer partitions . Given a partition λ = ( λ 1 , λ 2 , ..., λ n ) , where λ 1 ≥ λ 2 ≥ ... ≥ λ n , and each λ j is a non-negative integer, the functions a ( λ 1 + n − 1 , λ 2 + n − 2 , … , λ n ) ( x 1 , x 2 , … , x n ) = det [ x 1 λ 1 + n − 1 x 2 λ 1 + n − 1 … x n λ 1 + n − 1 x 1 λ 2 + n − 2 x 2 λ 2 + n − 2 … x n λ 2 + n − 2 ⋮ ⋮ ⋱ ⋮ x 1 λ n x 2 λ n … x n λ n ] {\displaystyle a_{(\lambda _{1}+n-1,\lambda _{2}+n-2,\dots ,\lambda _{n})}(x_{1},x_{2},\dots ,x_{n})=\det \left[{\begin{matrix}x_{1}^{\lambda _{1}+n-1}&x_{2}^{\lambda _{1}+n-1}&\dots &x_{n}^{\lambda _{1}+n-1}\\x_{1}^{\lambda _{2}+n-2}&x_{2}^{\lambda _{2}+n-2}&\dots &x_{n}^{\lambda _{2}+n-2}\\\vdots &\vdots &\ddots &\vdots \\x_{1}^{\lambda _{n}}&x_{2}^{\lambda _{n}}&\dots &x_{n}^{\lambda _{n}}\end{matrix}}\right]} are alternating polynomials by properties of

130-425: A ( n − 1 , n − 2 , … , 0 ) ( x 1 , x 2 , … , x n ) . {\displaystyle s_{\lambda }(x_{1},x_{2},\dots ,x_{n})={\frac {a_{(\lambda _{1}+n-1,\lambda _{2}+n-2,\dots ,\lambda _{n}+0)}(x_{1},x_{2},\dots ,x_{n})}{a_{(n-1,n-2,\dots ,0)}(x_{1},x_{2},\dots ,x_{n})}}.} This

195-474: A Gröbner basis for an appropriate elimination order. For example, is obviously a symmetric polynomial which is homogeneous of degree four, and we have The Schur polynomials occur in the representation theory of the symmetric groups , general linear groups , and unitary groups . The Weyl character formula implies that the Schur polynomials are the characters of finite-dimensional irreducible representations of

260-470: A different relation; see Romagny. From the perspective of representation theory , the symmetric and alternating polynomials are subrepresentations of the action of the symmetric group on n letters on the polynomial ring in n variables. (Formally, the symmetric group acts on n letters, and thus acts on derived objects, particularly free objects on n letters, such as the ring of polynomials.) The symmetric group has two 1-dimensional representations:

325-408: A number of branches of mathematics and physics , including the study of symmetric polynomials and of the symmetric group and in group representation theory in general. The seven partitions of 5 are Some authors treat a partition as a decreasing sequence of summands, rather than an expression with plus signs. For example, the partition 2 + 2 + 1 might instead be written as

390-400: A partition λ = ( λ 1 , λ 2 , ..., λ n ) , the Schur polynomial is a sum of monomials, where the summation is over all semistandard Young tableaux T of shape λ . The exponents t 1 , ..., t n give the weight of T , in other words each t i counts the occurrences of the number i in T . This can be shown to be equivalent to the definition from

455-423: A partition of k parts with largest part λ k {\displaystyle \lambda _{k}} . This statistic (which is unrelated to the one described above) appears in the study of Ramanujan congruences . There is a natural partial order on partitions given by inclusion of Young diagrams. This partially ordered set is known as Young's lattice . The lattice was originally defined in

520-503: A permutation module is decomposed into irreducible representations. There are several approaches to prove Schur positivity of a given symmetric function F . If F is described in a combinatorial manner, a direct approach is to produce a bijection with semi-standard Young tableaux. The Edelman–Greene correspondence and the Robinson–Schensted–Knuth correspondence are examples of such bijections. A bijection with more structure

585-454: A type of shape made by adjacent squares joined together, Young diagrams are a special kind of polyomino . The partition function p ( n ) {\displaystyle p(n)} counts the partitions of a non-negative integer n {\displaystyle n} . For instance, p ( 4 ) = 5 {\displaystyle p(4)=5} because the integer 4 {\displaystyle 4} has

650-441: Is No closed-form expression for the partition function is known, but it has both asymptotic expansions that accurately approximate it and recurrence relations by which it can be calculated exactly. It grows as an exponential function of the square root of its argument., as follows: In 1937, Hans Rademacher found a way to represent the partition function p ( n ) {\displaystyle p(n)} by

715-449: Is Giambelli's formula , which expresses the Schur function for an arbitrary partition in terms of those for the hook partitions contained within the Young diagram. In Frobenius' notation, the partition is denoted where, for each diagonal element in position ii , a i denotes the number of boxes to the right in the same row and b i denotes the number of boxes beneath it in

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780-589: Is a polynomial f ( x 1 , … , x n ) {\displaystyle f(x_{1},\dots ,x_{n})} such that if one switches any two of the variables, the polynomial changes sign: Equivalently, if one permutes the variables, the polynomial changes in value by the sign of the permutation : More generally, a polynomial f ( x 1 , … , x n , y 1 , … , y t ) {\displaystyle f(x_{1},\dots ,x_{n},y_{1},\dots ,y_{t})}

845-575: Is a proof using so called crystals . This method can be described as defining a certain graph structure described with local rules on the underlying combinatorial objects. A similar idea is the notion of dual equivalence. This approach also uses a graph structure, but on the objects representing the expansion in the fundamental quasisymmetric basis. It is closely related to the RSK-correspondence. Skew Schur functions s λ/μ depend on two partitions λ and μ, and can be defined by

910-433: Is a symmetric polynomial, the discriminant . That is, the ring of symmetric and alternating polynomials is a quadratic extension of the ring of symmetric polynomials, where one has adjoined a square root of the discriminant. Alternatively, it is: If 2 is not invertible, the situation is somewhat different, and one must use a different polynomial W n {\displaystyle W_{n}} , and obtains

975-409: Is a way of writing n as a sum of positive integers . Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition .) For example, 4 can be partitioned in five distinct ways: The only partition of zero is the empty sum, having no parts. The order-dependent composition 1 + 3 is the same partition as 3 + 1 , and

1040-702: Is equal to the number of Littlewood–Richardson tableaux of skew shape ν / λ {\displaystyle \nu /\lambda } and of weight μ {\displaystyle \mu } . Pieri's formula is a special case of the Littlewood-Richardson rule, which expresses the product h r s λ {\displaystyle h_{r}s_{\lambda }} in terms of Schur polynomials. The dual version expresses e r s λ {\displaystyle e_{r}s_{\lambda }} in terms of Schur polynomials. Evaluating

1105-408: Is given by The pentagonal number theorem gives a recurrence for q : where a k is (−1) if k = 3 m − m for some integer m and is 0 otherwise. By taking conjugates, the number p k ( n ) of partitions of n into exactly k parts is equal to the number of partitions of n in which the largest part has size k . The function p k ( n ) satisfies

1170-556: Is known as the bialternant formula of Jacobi . It is a special case of the Weyl character formula . This is a symmetric function because the numerator and denominator are both alternating, and a polynomial since all alternating polynomials are divisible by the Vandermonde determinant. The degree d Schur polynomials in n variables are a linear basis for the space of homogeneous degree d symmetric polynomials in n variables. For

1235-459: Is over all partitions μ such that μ / λ is a rim-hook of size r and ht ( μ / λ ) is the number of rows in the diagram μ / λ . The Littlewood–Richardson coefficients depend on three partitions , say λ , μ , ν {\displaystyle \lambda ,\mu ,\nu } , of which λ {\displaystyle \lambda } and μ {\displaystyle \mu } describe

1300-417: Is related to the generating function of p ( N , M ; n ) by the equality ∑ n = 0 M N p ( N , M ; n ) q n = ( M + N M ) q . {\displaystyle \sum _{n=0}^{MN}p(N,M;n)q^{n}={M+N \choose M}_{q}.} The rank of a partition is the largest number k such that

1365-402: Is said to be alternating in x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} if it changes sign if one switches any two of the x i {\displaystyle x_{i}} , leaving the y j {\displaystyle y_{j}} fixed. Products of symmetric and alternating polynomials (in

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1430-499: Is the Dedekind sum . The multiplicative inverse of its generating function is the Euler function ; by Euler's pentagonal number theorem this function is an alternating sum of pentagonal number powers of its argument. Srinivasa Ramanujan discovered that the partition function has nontrivial patterns in modular arithmetic , now known as Ramanujan's congruences . For instance, whenever

1495-438: Is the Vandermonde determinant a ( 2 , 1 , 0 ) ( x 1 , x 2 , x 3 ) {\displaystyle a_{(2,1,0)}(x_{1},x_{2},x_{3})} . Summarizing: Every homogeneous degree-four symmetric polynomial in three variables can be expressed as a unique linear combination of these four Schur polynomials, and this combination can again be found using

1560-843: Is the number of 1's, m 2 is the number of 2's, etc. (Components with m i = 0 may be omitted.) For example, in this notation, the partitions of 5 are written 5 1 , 1 1 4 1 , 2 1 3 1 , 1 2 3 1 , 1 1 2 2 , 1 3 2 1 {\displaystyle 5^{1},1^{1}4^{1},2^{1}3^{1},1^{2}3^{1},1^{1}2^{2},1^{3}2^{1}} , and 1 5 {\displaystyle 1^{5}} . There are two common diagrammatic methods to represent partitions: as Ferrers diagrams, named after Norman Macleod Ferrers , and as Young diagrams, named after Alfred Young . Both have several possible conventions; here, we use English notation , with diagrams aligned in

1625-688: Is the same as the number of partitions with distinct odd parts. Proof (outline) : The crucial observation is that every odd part can be " folded " in the middle to form a self-conjugate diagram: One can then obtain a bijection between the set of partitions with distinct odd parts and the set of self-conjugate partitions, as illustrated by the following example: [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] Among

1690-400: Is the subgroup of permutations such that λ w ( i ) = λ i {\displaystyle \lambda _{w(i)}=\lambda _{i}} for all i , and w acts on variables by permuting indices. The Murnaghan–Nakayama rule expresses a product of a power-sum symmetric function with a Schur polynomial, in terms of Schur polynomials: where the sum

1755-462: The Durfee square : The Durfee square has applications within combinatorics in the proofs of various partition identities. It also has some practical significance in the form of the h-index . A different statistic is also sometimes called the rank of a partition (or Dyson rank), namely, the difference λ k − k {\displaystyle \lambda _{k}-k} for

1820-539: The Murnaghan–Nakayama rule . Due to the connection with representation theory, a symmetric function which expands positively in Schur functions are of particular interest. For example, the skew Schur functions expand positively in the ordinary Schur functions, and the coefficients are Littlewood–Richardson coefficients. A special case of this is the expansion of the complete homogeneous symmetric functions h λ in Schur functions. This decomposition reflects how

1885-453: The characteristic of the coefficient ring is 2, there is no difference between the two concepts: the alternating polynomials are precisely the symmetric polynomials. The basic alternating polynomial is the Vandermonde polynomial : This is clearly alternating, as switching two variables changes the sign of one term and does not change the others. The alternating polynomials are exactly

1950-504: The complete symmetric functions and elementary symmetric functions , respectively. If the sum is taken over products of Schur polynomials in n {\displaystyle n} variables ( x 1 , … , x n ) {\displaystyle (x_{1},\dots ,x_{n})} , the sum includes only partitions of length ℓ ( λ ) ≤ n {\displaystyle \ell (\lambda )\leq n} since otherwise

2015-1202: The convergent series p ( n ) = 1 π 2 ∑ k = 1 ∞ A k ( n ) k ⋅ d d n ( 1 n − 1 24 sinh ⁡ [ π k 2 3 ( n − 1 24 ) ] ) {\displaystyle p(n)={\frac {1}{\pi {\sqrt {2}}}}\sum _{k=1}^{\infty }A_{k}(n){\sqrt {k}}\cdot {\frac {d}{dn}}\left({{\frac {1}{\sqrt {n-{\frac {1}{24}}}}}\sinh \left[{{\frac {\pi }{k}}{\sqrt {{\frac {2}{3}}\left(n-{\frac {1}{24}}\right)}}}\,\,\,\right]}\right)} where A k ( n ) = ∑ 0 ≤ m < k , ( m , k ) = 1 e π i ( s ( m , k ) − 2 n m / k ) . {\displaystyle A_{k}(n)=\sum _{0\leq m<k,\;(m,k)=1}e^{\pi i\left(s(m,k)-2nm/k\right)}.} and s ( m , k ) {\displaystyle s(m,k)}

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2080-1417: The determinant . A polynomial is alternating if it changes sign under any transposition of the variables. Since they are alternating, they are all divisible by the Vandermonde determinant a ( n − 1 , n − 2 , … , 0 ) ( x 1 , x 2 , … , x n ) = det [ x 1 n − 1 x 2 n − 1 … x n n − 1 x 1 n − 2 x 2 n − 2 … x n n − 2 ⋮ ⋮ ⋱ ⋮ 1 1 … 1 ] = ∏ 1 ≤ j < k ≤ n ( x j − x k ) . {\displaystyle a_{(n-1,n-2,\dots ,0)}(x_{1},x_{2},\dots ,x_{n})=\det \left[{\begin{matrix}x_{1}^{n-1}&x_{2}^{n-1}&\dots &x_{n}^{n-1}\\x_{1}^{n-2}&x_{2}^{n-2}&\dots &x_{n}^{n-2}\\\vdots &\vdots &\ddots &\vdots \\1&1&\dots &1\end{matrix}}\right]=\prod _{1\leq j<k\leq n}(x_{j}-x_{k}).} The Schur polynomials are defined as

2145-514: The first Giambelli formula using the Lindström–Gessel–Viennot lemma (as outlined on that page). Schur polynomials can be expressed as linear combinations of monomial symmetric functions m μ with non-negative integer coefficients K λμ called Kostka numbers , The Kostka numbers K λμ are given by the number of semi-standard Young tableaux of shape λ and weight μ . The first Jacobi−Trudi formula expresses

2210-408: The tuple (2, 2, 1) or in the even more compact form (2 , 1) where the superscript indicates the number of repetitions of a part. This multiplicity notation for a partition can be written alternatively as 1 m 1 2 m 2 3 m 3 ⋯ {\displaystyle 1^{m_{1}}2^{m_{2}}3^{m_{3}}\cdots } , where m 1

2275-399: The 22 partitions of the number 8, there are 6 that contain only odd parts : Alternatively, we could count partitions in which no number occurs more than once. Such a partition is called a partition with distinct parts . If we count the partitions of 8 with distinct parts, we also obtain 6: This is a general property. For each positive number, the number of partitions with odd parts equals

2340-550: The Ferrers diagram for the same partition is While this seemingly trivial variation does not appear worthy of separate mention, Young diagrams turn out to be extremely useful in the study of symmetric functions and group representation theory : filling the boxes of Young diagrams with numbers (or sometimes more complicated objects) obeying various rules leads to a family of objects called Young tableaux , and these tableaux have combinatorial and representation-theoretic significance. As

2405-521: The Schur functions being multiplied, and ν {\displaystyle \nu } gives the Schur function of which this is the coefficient in the linear combination; in other words they are the coefficients c λ , μ ν {\displaystyle c_{\lambda ,\mu }^{\nu }} such that The Littlewood–Richardson rule states that c λ , μ ν {\displaystyle c_{\lambda ,\mu }^{\nu }}

2470-680: The Schur polynomial s λ in (1, 1, ..., 1) gives the number of semi-standard Young tableaux of shape λ with entries in 1, 2, ..., n . One can show, by using the Weyl character formula for example, that s λ ( 1 , 1 , … , 1 ) = ∏ 1 ≤ i < j ≤ n λ i − λ j + j − i j − i . {\displaystyle s_{\lambda }(1,1,\dots ,1)=\prod _{1\leq i<j\leq n}{\frac {\lambda _{i}-\lambda _{j}+j-i}{j-i}}.} In this formula, λ ,

2535-481: The Schur polynomial as a determinant in terms of the complete homogeneous symmetric polynomials , where h i  := s ( i ) . The second Jacobi-Trudi formula expresses the Schur polynomial as a determinant in terms of the elementary symmetric polynomials , where e i  := s (1) and λ' is the conjugate partition to λ . In both identities, functions with negative subscripts are defined to be zero. Another determinantal identity

2600-445: The Schur polynomials vanish. There are many generalizations of these identities to other families of symmetric functions. For example, Macdonald polynomials, Schubert polynomials and Grothendieck polynomials admit Cauchy-like identities. The Schur polynomial can also be computed via a specialization of a formula for Hall–Littlewood polynomials , where S n λ {\displaystyle S_{n}^{\lambda }}

2665-741: The Vandermonde polynomial is a polynomial. Schur polynomials are defined in this way, as an alternating polynomial divided by the Vandermonde polynomial. Thus, denoting the ring of symmetric polynomials by Λ n , the ring of symmetric and alternating polynomials is Λ n [ v n ] {\displaystyle \Lambda _{n}[v_{n}]} , or more precisely Λ n [ v n ] / ⟨ v n 2 − Δ ⟩ {\displaystyle \Lambda _{n}[v_{n}]/\langle v_{n}^{2}-\Delta \rangle } , where Δ = v n 2 {\displaystyle \Delta =v_{n}^{2}}

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2730-420: The Vandermonde polynomial times a symmetric polynomial: a = v n ⋅ s {\displaystyle a=v_{n}\cdot s} where s {\displaystyle s} is symmetric. This is because: Conversely, the ratio of two alternating polynomials is a symmetric function, possibly rational (not necessarily a polynomial), though the ratio of an alternating polynomial over

2795-647: The analysis is more complicated. If n > 2 {\displaystyle n>2} , there are also other subrepresentations of the action of the symmetric group on the ring of polynomials, as discussed in representation theory of the symmetric group . Alternating polynomials are an unstable phenomenon: the ring of symmetric polynomials in n variables can be obtained from the ring of symmetric polynomials in arbitrarily many variables by evaluating all variables above x n {\displaystyle x_{n}} to zero: symmetric polynomials are thus stable or compatibly defined. However, this

2860-424: The character of the representation of the symmetric group indexed by the partition λ evaluated at elements of cycle type indexed by the partition ρ, then where ρ = (1, 2, 3, ...) means that the partition ρ has r k parts of length k . A proof of this can be found in R. Stanley's Enumerative Combinatorics Volume 2, Corollary 7.17.5. The integers χ ρ can be computed using

2925-434: The context of representation theory , where it is used to describe the irreducible representations of symmetric groups S n for all n , together with their branching properties, in characteristic zero. It also has received significant study for its purely combinatorial properties; notably, it is the motivating example of a differential poset . Alternating polynomials In algebra, an alternating polynomial

2990-495: The decimal representation of n {\displaystyle n} ends in the digit 4 or 9, the number of partitions of n {\displaystyle n} will be divisible by 5. In both combinatorics and number theory, families of partitions subject to various restrictions are often studied. This section surveys a few such restrictions. If we flip the diagram of the partition 6 + 4 + 3 + 1 along its main diagonal, we obtain another partition of 14: By turning

3055-555: The five partitions 1 + 1 + 1 + 1 {\displaystyle 1+1+1+1} , 1 + 1 + 2 {\displaystyle 1+1+2} , 1 + 3 {\displaystyle 1+3} , 2 + 2 {\displaystyle 2+2} , and 4 {\displaystyle 4} . The values of this function for n = 0 , 1 , 2 , … {\displaystyle n=0,1,2,\dots } are: The generating function of p {\displaystyle p}

3120-459: The general linear groups, and helps to generalize Schur's work to other compact and semisimple Lie groups . Several expressions arise for this relation, one of the most important being the expansion of the Schur functions s λ in terms of the symmetric power functions p k = ∑ i x i k {\displaystyle p_{k}=\sum _{i}x_{i}^{k}} . If we write χ ρ for

3185-737: The number and size of the parts. Let p ( N , M ; n ) denote the number of partitions of n with at most M parts, each of size at most N . Equivalently, these are the partitions whose Young diagram fits inside an M × N rectangle. There is a recurrence relation p ( N , M ; n ) = p ( N , M − 1 ; n ) + p ( N − 1 , M ; n − M ) {\displaystyle p(N,M;n)=p(N,M-1;n)+p(N-1,M;n-M)} obtained by observing that p ( N , M ; n ) − p ( N , M − 1 ; n ) {\displaystyle p(N,M;n)-p(N,M-1;n)} counts

3250-481: The number of partitions with distinct parts, denoted by q ( n ). This result was proved by Leonhard Euler in 1748 and later was generalized as Glaisher's theorem . For every type of restricted partition there is a corresponding function for the number of partitions satisfying the given restriction. An important example is q ( n ) (partitions into distinct parts). The first few values of q ( n ) are (starting with q (0)=1): The generating function for q ( n )

3315-410: The partition contains at least k parts of size at least k . For example, the partition 4 + 3 + 3 + 2 + 1 + 1 has rank 3 because it contains 3 parts that are ≥ 3, but does not contain 4 parts that are ≥ 4. In the Ferrers diagram or Young diagram of a partition of rank r , the r × r square of entries in the upper-left is known as

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3380-939: The partitions of n into exactly M parts of size at most N , and subtracting 1 from each part of such a partition yields a partition of n − M into at most M parts. The Gaussian binomial coefficient is defined as: ( k + ℓ ℓ ) q = ( k + ℓ k ) q = ∏ j = 1 k + ℓ ( 1 − q j ) ∏ j = 1 k ( 1 − q j ) ∏ j = 1 ℓ ( 1 − q j ) . {\displaystyle {k+\ell \choose \ell }_{q}={k+\ell \choose k}_{q}={\frac {\prod _{j=1}^{k+\ell }(1-q^{j})}{\prod _{j=1}^{k}(1-q^{j})\prod _{j=1}^{\ell }(1-q^{j})}}.} The Gaussian binomial coefficient

3445-457: The property Here, the inner product is the Hall inner product, for which the Schur polynomials form an orthonormal basis. Similar to the ordinary Schur polynomials, there are numerous ways to compute these. The corresponding Jacobi-Trudi identities are Integer partition In number theory and combinatorics , a partition of a non-negative integer n , also called an integer partition ,

3510-435: The ratio s λ ( x 1 , x 2 , … , x n ) = a ( λ 1 + n − 1 , λ 2 + n − 2 , … , λ n + 0 ) ( x 1 , x 2 , … , x n )

3575-505: The recurrence with initial values p 0 (0) = 1 and p k ( n ) = 0 if n ≤ 0 or k ≤ 0 and n and k are not both zero. One recovers the function p ( n ) by One possible generating function for such partitions, taking k fixed and n variable, is More generally, if T is a set of positive integers then the number of partitions of n , all of whose parts belong to T , has generating function This can be used to solve change-making problems (where

3640-602: The rows into columns, we obtain the partition 4 + 3 + 3 + 2 + 1 + 1 of the number 14. Such partitions are said to be conjugate of one another. In the case of the number 4, partitions 4 and 1 + 1 + 1 + 1 are conjugate pairs, and partitions 3 + 1 and 2 + 1 + 1 are conjugate of each other. Of particular interest are partitions, such as 2 + 2, which have themselves as conjugate. Such partitions are said to be self-conjugate . Claim : The number of self-conjugate partitions

3705-561: The same column (the arm and leg lengths, respectively). The Giambelli identity expresses the Schur function corresponding to this partition as the determinant of those for hook partitions. The Cauchy identity for Schur functions (now in infinitely many variables), and its dual state that and where the sum is taken over all partitions λ , and h λ ( x ) {\displaystyle h_{\lambda }(x)} , e λ ( x ) {\displaystyle e_{\lambda }(x)} denote

3770-488: The same variables x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} ) behave thus: This is exactly the addition table for parity , with "symmetric" corresponding to "even" and "alternating" corresponding to "odd". Thus, the direct sum of the spaces of symmetric and alternating polynomials forms a superalgebra (a Z 2 {\displaystyle \mathbf {Z} _{2}} - graded algebra ), where

3835-445: The set T specifies the available coins). As two particular cases, one has that the number of partitions of n in which all parts are 1 or 2 (or, equivalently, the number of partitions of n into 1 or 2 parts) is and the number of partitions of n in which all parts are 1, 2 or 3 (or, equivalently, the number of partitions of n into at most three parts) is the nearest integer to ( n + 3) / 12. One may also simultaneously limit

3900-409: The size of a part of the partition. The diagrams for the 5 partitions of the number 4 are shown below: An alternative visual representation of an integer partition is its Young diagram (often also called a Ferrers diagram). Rather than representing a partition with dots, as in the Ferrers diagram, the Young diagram uses boxes or squares. Thus, the Young diagram for the partition 5 + 4 + 1 is while

3965-441: The symmetric polynomials are the even part, and the alternating polynomials are the odd part. This grading is unrelated to the grading of polynomials by degree . In particular, alternating polynomials form a module over the algebra of symmetric polynomials (the odd part of a superalgebra is a module over the even part); in fact it is a free module of rank 1, with the Vandermonde polynomial in n variables as generator. If

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4030-443: The trivial representation and the sign representation. The symmetric polynomials are the trivial representation, and the alternating polynomials are the sign representation. Formally, the scalar span of any symmetric (resp., alternating) polynomial is a trivial (resp., sign) representation of the symmetric group, and multiplying the polynomials tensors the representations. In characteristic 2, these are not distinct representations, and

4095-606: The tuple indicating the width of each row of the Young diagram, is implicitly extended with zeros until it has length n . The sum of the elements λ i is d . See also the Hook length formula which computes the same quantity for fixed λ . The following extended example should help clarify these ideas. Consider the case n = 3, d = 4. Using Ferrers diagrams or some other method, we find that there are just four partitions of 4 into at most three parts. We have and so on, where Δ {\displaystyle \Delta }

4160-435: The two distinct compositions 1 + 2 + 1 and 1 + 1 + 2 represent the same partition as 2 + 1 + 1 . An individual summand in a partition is called a part . The number of partitions of n is given by the partition function p ( n ) . So p (4) = 5 . The notation λ ⊢ n means that λ is a partition of n . Partitions can be graphically visualized with Young diagrams or Ferrers diagrams . They occur in

4225-430: The upper-left corner. The partition 6 + 4 + 3 + 1 of the number 14 can be represented by the following diagram: [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] The 14 circles are lined up in 4 rows, each having

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