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| [[File:Semistandard_Young_tableaux_of_shape_(3,_2)_and_weight_(1,_1,_2,_1).png|thumb|right|The three semistandard Young tableaux of shape λ = (3, 2) and weight μ = (1, 1, 2, 1). They are counted by the Kostka number ''K''<sub>λμ</sub> = 3.]]
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| In [[mathematics]], the '''Kostka number''' ''K''<sub>λμ</sub> (depending on two [[Partition (number theory)|integer partitions]] λ and μ) is a [[non-negative integer]] that is equal to the number of [[semistandard Young tableaux]] of shape λ and weight μ. They were introduced by the mathematician [[Carl Kostka]] in his study of symmetric functions ({{harvtxt|Kostka|1882}}).<ref>Stanley, Enumerative combinatorics, volume 2, p. 398.</ref>
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| For example, if λ = (3, 2) and μ = (1, 1, 2, 1), the Kostka number ''K''<sub>λμ</sub> counts the number of ways to fill a left-aligned collection of boxes with 3 in the first row and 2 in the second row with 1 copy of the number 1, 1 copy of the number 2, 2 copies of the number 3 and 1 copy of the number 4 such that the entries increase along columns and do not decrease along rows. The three such tableaux are shown at right, and ''K''<sub>(3, 2) (1, 1, 2, 1)</sub> = 3.
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| ==Examples and special cases==
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| For any partition λ, the Kostka number ''K''<sub>λλ</sub> is equal to 1: the unique way to fill the [[Young diagram]] of shape λ = (λ<sub>1</sub>, λ<sub>2</sub>, ..., λ<sub>''m''</sub>) with λ<sub>1</sub> copies of 1, λ<sub>2</sub> copies of 2, and so on, so that the resulting tableau is weakly increasing along rows and strictly increasing along columns is if all the 1s are placed in the first row, all the 2s are placed in the second row, and so on. (This tableau is sometimes called the [[Yamanouchi tableau]] of shape λ.)
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| The Kostka number ''K''<sub>λμ</sub> is positive (i.e., there exist semistandard Young tableaux of shape λ and weight μ) if and only if λ and μ are both partitions of the same integer ''n'' and λ is larger than μ in [[dominance order]].<ref>Stanley, Enumerative combinatorics, volume 2, p. 315.</ref>
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| In general, there are no nice formulas known for the Kostka numbers. However, some special cases are known. For example, if μ = (1, 1, 1, ..., 1) is the partition whose parts are all 1 then a semistandard Young tableau of weight μ is a standard Young tableau; the number of standard Young tableaux of a given shape λ is given by the [[hook-length formula]].
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| ==Kostka numbers, symmetric functions and representation theory==
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| In addition to the purely [[combinatorics|combinatorial]] definition above, they can also be defined as the coefficients that arise when one expresses the [[Schur polynomial]] ''s''<sub>λ</sub> as a [[linear combination]] of [[Ring_of_symmetric_functions#Defining_individual_symmetric_functions|monomial symmetric functions]] ''m''<sub>μ</sub>:
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| : <math>s_\lambda= \sum_\mu K_{\lambda\mu}m_\mu.\ </math>
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| Because of the connections between symmetric function theory and [[representation theory]], Kostka numbers also express the decomposition of the [[permutation module]] ''M''<sub>μ</sub> in terms of the representations ''V''<sub>λ</sub> corresponding to the character ''s''<sub>λ</sub>, i.e.,
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| : <math>M_\mu = \bigoplus_{\lambda} K_{\lambda \mu} V_\lambda.</math>
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| On the level of representations of the [[general linear group]] <math>\mathrm{GL}_n(\mathbb{C})</math>, the Kostka number ''K''<sub>λμ</sub> counts the dimension of the [[weight space]] corresponding to μ in the [[irreducible representation]] ''V''<sub>λ</sub> (where we require μ and λ to have at most ''n'' parts).
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| ==Examples==
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| The Kostka numbers for partitions of size at most 3 are as follows:
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| : ''K''<sub>(0) (0)</sub> = 1 (here (0) represents the empty partition)
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| : ''K''<sub>(1) (1)</sub> = 1
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| : ''K''<sub>(2) (2)</sub> = ''K''<sub>(2) (1,1)</sub> = ''K''<sub>(1,1) (1,1)</sub> = 1, ''K''<sub>(1,1) (2)</sub> = 0.
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| : ''K''<sub>(3) (3)</sub> = ''K''<sub>(3) (2,1)</sub> = ''K''<sub>(3) (2,1)</sub> = 1
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| : ''K''<sub>(2,1) (3)</sub> = 0, ''K''<sub>(2,1) (2,1)</sub> = 1, ''K''<sub>(2,1) (1,1,1)</sub> = 2
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| : ''K''<sub>(1,1,1) (3)</sub> = ''K''<sub>(1,1,1) (2,1)</sub> = 0, ''K''<sub>(1,1,1) (1,1,1)</sub> = 1
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| These values are exactly the coefficients in the expansions of Schur functions in terms of monomial symmetric functions:
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| :''s'' = ''m'' = 1 (indexed by the empty partition)
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| :''s''<sub>1</sub> = ''m''<sub>1</sub>
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| :''s''<sub>2</sub> = ''m''<sub>2</sub> + ''m''<sub>11</sub>
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| :''s''<sub>11</sub> = ''m''<sub>11</sub>
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| :''s''<sub>3</sub> = ''m''<sub>3</sub> + ''m''<sub>21</sub> + ''m''<sub>111</sub>
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| :''s''<sub>21</sub> = ''m''<sub>21</sub> + 2''m''<sub>111</sub>
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| :''s''<sub>111</sub> = ''m''<sub>111.</sub>
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| {{harvtxt|Kostka|1882|loc=pages 118-120}} gave tables of these numbers for partitions of numbers up to 8.
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| ==Generalizations==
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| Kostka numbers are special values of the 1 or 2 variable [[Kostka polynomial]]s:
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| : <math>K_{\lambda\mu}= K_{\lambda\mu}(1)=K_{\lambda\mu}(0,1). </math>
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{citation|authorlink=Richard P. Stanley | first = Richard |last=Stanley|title=Enumerative combinatorics, volume 2|year=1999|publisher=Cambridge University Press}}
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| *{{citation|authorlink=C. Kostka | first=C. |last=Kostka|title=Über den Zusammenhang zwischen einigen Formen von symmetrischen Funktionen|journal= [[Crelle's Journal]]|volume= 93 |year=1882|pages=89–123|url=http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=259790}}
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| *{{Citation | last1=Macdonald | first1=I. G. | author1-link=Ian G. Macdonald | title=Symmetric functions and Hall polynomials | url=http://www.oup.com/uk/catalogue/?ci=9780198504504 | publisher=The Clarendon Press Oxford University Press | edition=2nd | series=Oxford Mathematical Monographs | isbn=978-0-19-853489-1 | id={{MathSciNet | id = 1354144}} | year=1995}}
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| *{{springer|id=s/s120040|title=Schur functions in algebraic combinatorics|first=Bruce E. |last=Sagan | authorlink=Bruce Sagan}}
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| [[Category:Symmetric functions]]
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| [[Category:Numbers]]
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