Augmented monomials in terms of power sums

The problem of base changes for the classical symmetric functions has been solved a long time ago and has been incorporated into most computer software packages for symmetric functions. In this paper, we develop a simple recursive formula for the expansion of the augmented monomial symmetric functions into power sum symmetric functions. As corollaries, we present two algorithms that can be used to expressing the augmented monomial symmetric functions in terms of the power sum symmetric functions.


Background
Any positive integer n can be written as a sum of one or more positive integers, i.e., When the order of integers i does not matter, this representation is known as an integer partition Andrews (1976) and can be rewritten as where each positive integer i appears t i times. If the order of integers i is important, then the representation (1) is known as a composition. For we have a descending composition. We notice that more often than not there appears the tendency of defining partitions as descending compositions and this is also the convention used in this paper. In order to indicate that is a partition of n, we use the notation ⊢ n. We denote by l( ) the number of parts of , i.e., (1) n = 1 + 2 + · · · + r . n = t 1 + 2t 2 + · · · + nt n 1 2 · · · r = [ 1 , 2 , . . . , r ] or = [1 t 1 2 t 2 . . . n t n ] l( ) = r or l( ) = t 1 + t 2 + · · · + t n .
If α, β ⊢ n, then α precedes β in the dominance order if and only if for any k 1, the sum of the k largest parts of α is less than the sum of the k largest parts of β, i.e., for all k 1. In this definition, partitions are extended by appending zero parts at the end as necessary. If α = [α 1 , α 2 , . . . , α r ] and β = [β 1 , β 2 , . . . , β s ] are partitions of the same positive integer, then α precedes β in the lexicographic order if there is a positive integer t with the following properties: 1. t r and t s; 2. for ever positive integer i t, α i = β i ; and 3. either α t+1 < β t+1 or t = r and t < s.
When α precedes β in lexicographic order, we use the notation α ≺ β. If α ≺ β or α = β, then we use the notation α β. It is clear that the dominance order implies lexicographical order.
We recall some basic facts about monomial symmetric functions. Proofs and details can be found in Macdonald's book (Macdonald 1995). Let = [ 1 , 2 , . . . , k ] be a partition with k n. Being given a set of variables {x 1 , x 2 , . . . , x n }, the monomial symmetric function on these variables is the sum of monomial x 1 1 x 2 2 , . . . , x k k and all distinct monomials obtained from it by a permutation of variables. For instance, with = [2, 1, 1] and n = 4 , we have: In particular, when = [k], we have the kth power sum symmetric function p k = p k (x 1 , x 2 , . . . , x n ), i.e., In every case p 0 (x 1 , x 2 , . . . , x n ) = n.
If ⊢ k then m is a symmetric function of degree k. It is well-known that the set is a basis for the vector space k n of symmetric functions of degree k of n variables. The dimension of k n is the number of partitions of k. The power sum symmetric functions p k do not have enough elements to form a basis for k n , there must be one function for every partition ⊢ k. To that end in each case we form multiplicative function p = p (x 1 , x 2 , . . . , x n ) so that for {m (x 1 , x 2 , . . . , x n ) | ⊢ k and l( ) n} we note Also, it is known that the set is another basis for k n . For each partition with k n, the augmented monomial symmetric function is defined by In this paper, we develop a simple recursive formula for the expansion of the augmented monomial symmetric functions into power sum symmetric functions. As corollaries, we present two algorithms that can be used to expressing the augmented monomial symmetric functions in terms of the power sum symmetric functions.

Two theorems for expanding augmented monomials
The cardinality of a set A is usually denoted |A|. Recall that a partition of the set A is a collection of non-empty, pairwise disjoint subsets of A whose union is A.
A simple way to express the augmented monomial symmetric function m in terms of the power sum is given by Proof We denote by M the set of terms in the expression p k ·m [ 1 , 2 ,..., k−1 ] , by M k the set of terms in the expression m [ 1 , 2 ,..., k ] and by M i the set of terms in the expression m [ 1 ,..., i−1 , i + k , i+1 ,..., k−1 ] , for i = 1, 2, . . . , k − 1. According to and p = p 1 p 2 · · · p l( ) .
Therefore, the theorem is proved.
Example 1 Replacing k by 2 in Theorem 1, we get Then, for k = 3, we obtain By (2) and (3), we deduce that It is clear that in the expansion of the augmented monomial m generated by Theorem 1, the number of terms is equal to the number of parts of .
The following result is immediate from Theorem 1.
Corollary 1 Let = [1 t 1 2 t 2 · · · ] be an integer partition and let j be a positive integer such that t j > 0. Then where δ ij is the Kronecker delta and with and for all i > 0.
In this corollary, if ⊢ k then we remark that 0 ⊢ k − j and ≺ i for all i > 0 with t i > δ ij . If t j = 1 then we have t j (j) = −1. This drawback is eliminated by the fact that t j − δ jj = 0.
Example 2 For = [1 3 2 1 3 1 ] and j = 3, by Corollary 1, we have Clearly, the coefficient of m 0 is p 3 , the coefficient of m 1 is −3, the coefficient of m 2 is −1 , and for i > 2 all the coefficients are 0. Thus, we obtain We remark that in the expansion of m generated by Corollary 1, the number of terms is equal to So, we can say that this corollary is a refined form of Theorem 1.
We denote by P n the set of all partitions of {1, 2, . . . , n}. The cardinality of the set P n is well-known as the nth Bell number, B n (see Sloane 2012, A000110). The Möbius function of P n (Bender and Goldman 1975;Rota 1964), namely can be used to express the augmented monomial symmetric functions in terms of the power sum symmetric functions. the number of distinct parts of + 1, for t j > 1, 0, for t j = 1.
Let P ′ k be a subset of P k defined by We are to prove the theorem by induction on k. For k = 1, we have µ({1}) = 1 and , the base case of induction is finished. We suppose that the relation is true for any integer k ′ , 1 k ′ < k. By (5), (6)

Iterative algorithm for computing transition matrix
If ⊢ k, then it is immediate from Theorem 1 or Theorem 2 the fact that the augmented monomial symmetric function m is a sum over integer partitions of k.

Corollary 2 Let be an integer partition. Then
where T β is an integer such that m and p are functions of n variables, n l( ).
We observe that the transition matrix expanding the augmented monomial symmetric functions in p is lower triangular (with respect to any extension of the dominance ordering on partitions to a total order on the partitions ⊢ k), i.e., where with Example 4 For k = 4, according to Theorems 1 or 2, we obtain We remark that where e k is the kth elementary symmetric function. For k = t 1 + 2t 2 + · · · + kt k , the number of ways of partitioning a set of k different objects into t i subsets containing i objects, i = 1, 2, . . . , k is [see (s.24.1.2 Abramovitz and Stegun 1972)]. Thus, the formula where k = t 1 + 2t 2 + · · · + kt k , can be easily derived from Theorem 2. Unfortunately, for T β with [1 k ] ≺ and ≺ β such formulas are not known.
The following result is immediate from Corollaries 1 and 2.
Corollary 3 Let k be a positive integer. If = [1 t 1 2 t 2 · · · ] and β = [1 v 1 2 v 2 · · · ] are two integer partitions of k such that ≺ β then where j is a positive integer such that t j > 0, δ ij is the Kronecker delta, with and for all i > 0.
In this corollary, for v j = 0 we have v j (0) = −1. Fortunately, this drawback is eliminated by the fact that 1 − δ 0,v j = 0. Recall that 0 is an integer partition of k − j and ≺ i for all i > 0 with t i > δ ij . We remark that β 0 ⊢ k − j for v j > 0.
Example 5 By Corollary 3, for = [1 4 ] and β = [1 1 3 1 ], we have According to (7) and Corollary 3, we obtain Algorithm 1 for computing the transition matrix T (k) . We can see that in order to compute the transition matrix T (k) , Algorithm 1 is based on generating the immediate lexicographic predecessor of an integer partition (see lines 10 and 22). The problem of generating the immediate lexicographic predecessor of an integer partition is well-known in literature. For more details, one can refer to (Kelleher and O'Sullivan 2009) and the references therein.  visit T (r)

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r ← r + 1 31: end while Example 6 Applying Algorithm 1 for k = 5, we get successively: if t i > 0 and t i < t j then