For example, if and , the Kostka number 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 .
Examples and special casesedit
For any partition , the Kostka number is equal to 1: the unique way to fill the Young diagram of shape with copies of 1, 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 .)
The Kostka number is positive (i.e., there exist semistandard Young tableaux of shape and weight ) if and only if and are both partitions of the same integer and is larger than in dominance order.[2]
In general, there are no nice formulas known for the Kostka numbers. However, some special cases are known. For example, if is the partition whose parts are all 1 then a semistandard Young tableau of weight is a standard Young tableau; the number of standard Young tableaux of a given shape is given by the hook-length formula.
Propertiesedit
An important simple property of Kostka numbers is that does not depend on the order of entries of . For example, . This is not immediately obvious from the definition but can be shown by establishing a bijection between the sets of semistandard Young tableaux of shape and weights and , where and differ only by swapping two entries.[3]
Kostka numbers, symmetric functions and representation theoryedit
The Kostka numbers for partitions of size at most 3 are as follows:
These values are exactly the coefficients in the expansions of Schur functions in terms of monomial symmetric functions:
Kostka (1882, pages 118-120) gave tables of these numbers for partitions of numbers up to 8.
Generalizationsedit
Kostka numbers are special values of the 1 or 2 variable Kostka polynomials:
Notesedit
^Stanley, Enumerative combinatorics, volume 2, p. 398.
^Stanley, Enumerative combinatorics, volume 2, p. 315.
^Stanley, Enumerative combinatorics, volume 2, p. 311.
^Stanley, Enumerative combinatorics, volume 2, p. 311.
Referencesedit
Stanley, Richard (1999), Enumerative combinatorics, volume 2, Cambridge University Press
Kostka, C. (1882), "Über den Zusammenhang zwischen einigen Formen von symmetrischen Funktionen", Crelle's Journal, 93: 89–123, doi:10.1515/crll.1882.93.89
Macdonald, I. G. (1995), Symmetric functions and Hall polynomials, Oxford Mathematical Monographs (2nd ed.), The Clarendon Press Oxford University Press, ISBN 978-0-19-853489-1, MR 1354144, archived from the original on 2012-12-11