Page 13 - Textos de Matemática Vol. 44

Z =
xβ1 xβn
per(c1(A), ..., c1(A), ..., cn(A), ..., cn(A)) 1 · · · n . β1! βn!
NATA´LIA BEBIANO’S LEGACY 3
i = 1,...,n. The product Z = z1z2 ···zn is hence a homogeneous polynomial of degree n in x1, . . . , xn. Nata´lia observes that if we apply the n-th totally mixed partial derivative to Z we obtain precisely the permanent:
aiσi.
By selecting the n columns c1(A), · · · , cn(A) of A from the left to the right possibly with multiplicities (that can be 0) in all possible ways, forming the permanents of these matrices and multiplying them with adequate monomials of degree n in x1, · · · , xn we can recover Z: More precisely,
∂ n n per(A)=∂x∂x ···∂x Z=
12 n σ∈Sni=1
β1,...,βn ≥0 β1 + · · · + βn = n
β1
βn
Using this formula for A′ = (aijxj) instead of A Nat´alia deduces Ryser’s formula for the evaluation of permanents without using the inclusion-exclusion principle. Let Qk,n be the set of integer k-tuples w = (w1,...wk) satisfying 1 ≤ w1 < w2 < ... < wk ≤ n, and denote by w¯ the complementary strictly increasing n − k-tuple. Denote by Aw the matrix obtained from A by replacing the columns with indices in w¯ by 0s and let S(Aw) be the product of the row sums of Aw. Then Ryser’s formula is
n
per(A) = (−1)n−kS(Aw).
k=1 w∈Qk,n
In the paper she extends the above formula also to get the following expres-
sion for the n-th power of a quadratic form:  n
1 n xα1 xα yβ1 yβ 1 nn1 nn
xiaijyj = per(A(α1,...,αn;β1,...,βn)) ··· ··· . n!  α1! αn! β1! βn!
i,j =1 α,β
Here αi = βj = n, and A(α1, ..., αn; β1, ..., βn) is the matrix obtained from A by replacing its rows by αi copies of row i of A, for i = 1,2,...,n, and then in the resulting matrix replacing the columns by βj copies of column j, j = 1,...,n. Minc’s formula of Binet type for the permanent also receives a new proof. In the follow up [B92] some of the results above are given proofs by means of differential operators this times with the help of multilinear algebra