Utilisateur:Darkpatric/Coefficient binomial de Gauss
En Mathématiques, le coefficient binomial de Gauss ( ou encore le coefficient de Gauss, ou coefficient q-binomial sont Q-analoguedes Coefficient binomiaux. Ils sont notés or , et sont des fonctions polynomiales à coefficients entiers en q. Lorsque l'on fixé q à la puissance d'un nombre premier, compte le nombre de sous espaces de dimension k dans un Espace vectoriel de dimension n sur un Corps fini à q éléments.
Définition
modifierLes coefficients binomiaux de Gauss sont définit par :
avec m et r des entiers positifs. Pour r=0 la valeur est 1 puisque le numérateur et le dénominateur sont tout deux des produits sur l'ensemble vide. D'autre part, bien que la formule semble en premier lieu impliquer une fonction rationnelle en q, elle désigne en réalité une fonction polynomiale car le reste de la division dans Z[q] est nul. Il faut remarquer que l'on peut appliquer la formule pour r = m + 1, cela donne 0 à cause du facteur 1 − q0 = 0 présent au numérateur, de manière cohérente avec le second cas
in accordance with the second clause (for even larger r the factor 0 remains present in the numerator, but its further factors would involve negative powers of q, whence explicitly stating the second clause is preferable). All of the factors in numerator and denominator are divisible by 1 − q, with as quotient a q number:
dividing out these factors gives the equivalent formula
which makes evident the fact that substituting q = 1 into gives the ordinary binomial coefficient In terms of the q factorial , the formula can be stated as
a compact form (often given as only definition), which however hides the presence of many common factors in numerator and denominator. This form does make obvious the symmetry for r ≤ m.
Unlike the ordinary binomial coefficient, the Gaussian binomial coefficient has finite values for (the limit being analytically meaningful for |q|<1):
Examples
modifierCombinatorial description
modifierInstead of these algebraic expressions, one can also give a combinatorial definition of Gaussian binomial coefficients. The ordinary binomial coefficient counts the r-combinations chosen from an m-element set. If one takes those m elements to be the different character positions in a word of length m, then each r-combination corresponds to a word of length m using an alphabet of two letters, say {0,1}, with r copies of the letter 1 (indicating the positions in the chosen combination) and m − r letters 0 (for the remaining positions).
The words using 0s and 1s would be 0011, 0101, 0110, 1001, 1010, 1100.
To obtain from this model the Gaussian binomial coefficient , it suffices to count each word with a factor qd, where d is the number of "inversions" of the word: the number of pairs of positions for which the leftmost position of the pair holds a letter 1 and the rightmost position holds a letter 0 in the word. For example, there is one word with 0 inversions, 0011. There is 1 with only a single inversion, 0101. There are two words with 2 inversions, 0110, and 1001. There is one with 3, 1010, and finally one word with 4 inversions, 1100. This corresponds to the coefficients in .
It can be shown that the polynomials so defined satisfy the Pascal identities given below, and therefore coincide with the polynomials given by the algebraic definitions. A visual way to view this definition is to associate to each word a path across a rectangular grid with sides of height r and width m − r, from the bottom left corner to the top right corner, taking a step right for each letter 0 and a step up for each letter 1. Then the number of inversions of the word equals the area of the part of the rectangle that is to the bottom-right of the path.
Properties
modifierLike the ordinary binomial coefficients, the Gaussian binomial coefficients are center-symmetric, i.e., invariant under the reflection :
In particular,
The name Gaussian binomial coefficient stems from the fact[réf. nécessaire] that their evaluation at q = 1 is
for all m and r.
The analogs of Pascal identities for the Gaussian binomial coefficients are
and
There are analogs of the binomial formula, and of Newton's generalized version of it for negative integer exponents, although for the former the Gaussian binomial coefficients themselves do not appear as coefficients:
and
which, for become:
and
The first Pascal identity allows one to compute the Gaussian binomial coefficients recursively (with respect to m ) using the initial "boundary" values
and also incidentally shows that the Gaussian binomial coefficients are indeed polynomials (in q). The second Pascal identity follows from the first using the substitution and the invariance of the Gaussian binomial coefficients under the reflection . Both Pascal identities together imply
which leads (when applied iteratively for m, m − 1, m − 2,....) to an expression for the Gaussian binomial coefficient as given in the definition above.
Applications
modifierGaussian binomial coefficients occur in the counting of symmetric polynomials and in the theory of partitions. The coefficient of qr in
is the number of partitions of r with m or fewer parts each less than or equal to n. Equivalently, it is also the number of partitions of r with n or fewer parts each less than or equal to m.
Gaussian binomial coefficients also play an important role in the enumerative theory of projective spaces defined over a finite field. In particular, for every finite field Fq with q elements, the Gaussian binomial coefficient
counts the number of k-dimensional vector subspaces of an n-dimensional vector space over Fq (a Grassmannian). When expanded as a polynomial in q, it yields the well-known decomposition of the Grassmannian into Schubert cells. For example, the Gaussian binomial coefficient
is the number of one-dimensional subspaces in (Fq)n (equivalently, the number of points in the associated projective space). Furthermore, when q is 1 (respectively −1), the Gaussian binomial coefficient yields the Euler characteristic of the corresponding complex (respectively real) Grassmannian.
The number of k-dimensional affine subspaces of Fqn is equal to
- .
This allows another interpretation of the identity
as counting the (r − 1)-dimensional subspaces of (m − 1)-dimensional projective space by fixing a hyperplane, counting such subspaces contained in that hyperplane, and then counting the subspaces not contained in the hyperplane; these latter subspaces are in bijective correspondence with the (r − 1)-dimensional affine subspaces of the space obtained by treating this fixed hyperplane as the hyperplane at infinity.
In the conventions common in applications to quantum groups, a slightly different definition is used; the quantum binomial coefficient there is
- .
This version of the quantum binomial coefficient is symmetric under exchange of and .
Triangles
modifierThe Gaussian binomial coefficients can be arranged in a triangle for each q, which is Pascal's triangle for q=1.
Read line by line these triangles form the following sequences in the OEIS:
References
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