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Augmented monomials in terms of power sums
SpringerPlus volume 4, Article number: 724 (2015)
Abstract
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 \(\lambda _i\) 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 \(\lambda _i\) 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 \(\lambda \vdash n\). We denote by \(l(\lambda )\) the number of parts of \(\lambda\), i.e.,
If \(\alpha ,\beta \vdash n\), then \(\alpha\) precedes \(\beta\) in the dominance order if and only if for any \(k\geqslant 1\), the sum of the k largest parts of \(\alpha\) is less than the sum of the k largest parts of \(\beta\), i.e.,
for all \(k\geqslant 1\). In this definition, partitions are extended by appending zero parts at the end as necessary. If \(\alpha =[\alpha _1,\alpha _2,\ldots ,\alpha _r]\) and \(\beta =[\beta _1,\beta _2,\ldots ,\beta _s]\) are partitions of the same positive integer, then \(\alpha\) precedes \(\beta\) in the lexicographic order if there is a positive integer t with the following properties:

1.
\(t\leqslant r\) and \(t\leqslant s\);

2.
for ever positive integer \(i\leqslant t\), \(\alpha _i=\beta _i\); and

3.
either \(\alpha _{t+1}<\beta _{t+1}\) or \(t=r\) and \(t<s\).
When \(\alpha\) precedes \(\beta\) in lexicographic order, we use the notation \(\alpha \prec \beta\). If \(\alpha \prec \beta\) or \(\alpha\) = \(\beta\), then we use the notation \(\alpha \preceq \beta\). 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 \(\lambda =[\lambda _1 ,\lambda _2 ,\ldots ,\lambda _k]\) be a partition with \(k\leqslant n\). Being given a set of variables \(\{x_1,x_2,\ldots ,x_n\}\), the monomial symmetric function
on these variables is the sum of monomial \(x_1^{\lambda _1}x_2^{\lambda _2},\ldots , x_k^{\lambda _k}\) and all distinct monomials obtained from it by a permutation of variables. For instance, with \(\lambda =[2,1,1]\) and \(n=4\), we have:
In particular, when \(\lambda =[k]\), we have the kth power sum symmetric function \(p_k=p_k (x_1,x_2,\dots ,x_n)\), i.e.,
In every case \(p_0(x_1,x_2,\ldots ,x_n)=n\).
If \(\lambda \vdash k\) then \(m_{\lambda }\) is a symmetric function of degree k. It is wellknown that the set
is a basis for the vector space \(\Lambda _n^k\) of symmetric functions of degree k of n variables. The dimension of \(\Lambda _n^k\) is the number of partitions of k. The power sum symmetric functions \(p_k\) do not have enough elements to form a basis for \(\Lambda _n^k\), there must be one function for every partition \(\lambda \vdash k\). To that end in each case we form multiplicative function \(p_{\lambda }=p_{\lambda }(x_1,x_2,\ldots ,x_n)\) so that for
we note
Also, it is known that the set
is another basis for \(\Lambda _n^k\).
For each partition
with \(k\leqslant n\), 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 \(\left A \right\). Recall that a partition of the set A is a collection of nonempty, pairwise disjoint subsets of A whose union is A.
A simple way to express the augmented monomial symmetric function \(\tilde{m}_{\lambda }\) in terms of the power sum is given by
Theorem 1
Let \([\lambda _1,\lambda _2,\ldots ,\lambda _k]\) be an integer partition. Then
where \(\tilde{m}\) and p are functions of n variables, \(n\geqslant k\).
Proof
We denote by M the set of terms in the expression \(p_{\lambda _k}\cdot \tilde{m}_{[\lambda _1,\lambda _2,\ldots ,\lambda _{k1}]}\), by \(M_k\) the set of terms in the expression \(\tilde{m}_{[\lambda _1,\lambda _2,\ldots ,\lambda _k]}\) and by \(M_i\) the set of terms in the expression \(\tilde{m}_{[\lambda _1,\ldots ,\lambda _{i1},\lambda _{i}+\lambda _{k},\lambda _{i+1},\ldots ,\lambda _{k1}]}\), for \(i=1,2,\ldots ,k1\). According to
and
we get
Taking into account that \(\{M_i\}_{1\leqslant i \leqslant k}\) is pairwise disjoint, we deduce that \(\{M_i\}_{1\leqslant i \leqslant k}\) is a set partition of M. Therefore, the theorem is proved. \(\square\)
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 \(\tilde{m}_{\lambda }\) generated by Theorem 1, the number of terms is equal to the number of parts of \(\lambda\).
The following result is immediate from Theorem 1.
Corollary 1
Let \(\lambda =[1^{t_1}2^{t_2}\cdots ]\) be an integer partition and let j be a positive integer such that \(t_j>0\) . Then
where \(\delta _{ij}\) is the Kronecker delta and
with
and
for all \(i>0\).
In this corollary, if \(\lambda \vdash k\) then we remark that \(\lambda ^0 \vdash kj\) and \(\lambda \prec \lambda ^i\) for all \(i>0\) with \(t_i>\delta _{ij}\). If \(t_j=1\) then we have \(t_j(j)=1\). This drawback is eliminated by the fact that \(t_j\delta _{jj}=0\).
Example 2
For \(\lambda =[1^32^13^1]\) and \(j=3\), by Corollary 1, we have
Clearly, the coefficient of \(\tilde{m}_{\lambda ^0}\) is \(p_3\), the coefficient of \(\tilde{m}_{\lambda ^1}\) is \(3\), the coefficient of \(\tilde{m}_{\lambda ^2}\) is \(1\), and for \(i>2\) all the coefficients are 0. Thus, we obtain
We remark that in the expansion of \(\tilde{m}_{\lambda }\) 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 \(\mathcal {P}_n\) the set of all partitions of \(\left\{ 1,2,\ldots ,n \right\}\). The cardinality of the set \(\mathcal {P}_n\) is wellknown as the nth Bell number, \(B_n\) (see Sloane 2012, A000110). The Möbius function of \(\mathcal {P}_n\) (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.
Theorem 2
Let \(\lambda\) be an integer partition. Then
where \(s(v)=[s_1,s_2,\ldots ,s_{\left v \right }]\) is an integer partition with
\(\tilde{m}\) and p are functions of n variables, \(n\geqslant l(\lambda )\).
Proof
Let \(\lambda =[\lambda _1,\lambda _2,\ldots ,\lambda _k]\) be an integer partition. For \(v=(v_1,v_2,\ldots ,v_r)\in \mathcal {P}_{k1}\) and \(1\leqslant i \leqslant r\) let us consider \(f(v),f_i(v) \in \mathcal {P}_k\) defined by
and
By (4), we deduce that
Let \(\mathcal {P'}_k\) be a subset of \(\mathcal {P}_k\) defined by
We are to prove the theorem by induction on k. For \(k=1\), we have \(\mu (\{1\})=1\) and \(s(\{1\})=[\lambda _1]\). Considering that \(\tilde{m}_{[\lambda _1]}=\mu (\{1\})p_{s(\{1\})}\), the base case of induction is finished. We suppose that the relation
is true for any integer \(k'\), \(1\leqslant k' < k\). By (5), (6) and Theorem 1, we can write
Thus, the proof is finished. \(\square\)
Example 3
For \(\{1,2,3\}\), we have \(\mathcal {P}_3=\{a,b,c,d,e\}\) with
According to (4), we have
Taking into account Theorem 2, we get
Iterative algorithm for computing transition matrix
If \(\lambda \vdash k\), then it is immediate from Theorem 1 or Theorem 2 the fact that the augmented monomial symmetric function \(\tilde{m}_{\lambda }\) is a sum over integer partitions of k.
Corollary 2
Let \(\lambda\) be an integer partition. Then
where \({T}_{\lambda \beta }\) is an integer such that
\(\tilde{m}\) and p are functions of n variables, \(n\geqslant l(\lambda )\).
We observe that the transition matrix expanding the augmented monomial symmetric functions in \(p_{\lambda }\) is lower triangular (with respect to any extension of the dominance ordering on partitions to a total order on the partitions \(\lambda \vdash k\)), 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 \(k=t_1+2t_2+\cdots +kt_k\), the number of ways of partitioning a set of k different objects into \(t_i\) subsets containing i objects, \(i=1,2,\ldots ,k\) is
[see (s.24.1.2 Abramovitz and Stegun 1972)]. Thus, the formula
where \(k=t_1+2t_2+\cdots +kt_k\), can be easily derived from Theorem 2. Unfortunately, for \(T_{\lambda \beta }\) with \([1^k]\prec \lambda\) and \(\lambda \prec \beta\) such formulas are not known.
The following result is immediate from Corollaries 1 and 2.
Corollary 3
Let k be a positive integer. If \(\lambda =[1^{t_1}2^{t_2}\cdots ]\) and \(\beta =[1^{v_1}2^{v_2}\cdots ]\) are two integer partitions of k such that \(\lambda \prec \beta\) then
where j is a positive integer such that \(t_j>0\), \(\delta _{ij}\) 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 \(1\delta _{0,v_j}=0\). Recall that \(\lambda ^0\) is an integer partition of \(kj\) and \(\lambda \prec \lambda ^i\) for all \(i>0\) with \(t_i>\delta _{ij}\). We remark that \(\beta ^0 \vdash kj\) for \(v_j>0\).
Example 5
By Corollary 3, for \(\lambda =[1^4]\) and \(\beta =[1^13^1]\), 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 wellknown in literature. For more details, one can refer to (Kelleher and O’Sullivan 2009) and the references therein.
Example 6
Applying Algorithm 1 for \(k=5\), we get successively:
At the end of this section, we remark the following
Conjecture 1
Let k be a positive integer. The identities
are true for all \(\lambda \prec [k]\).
Recursive algorithm for computing an element of the transition matrix
A specific augmented monomial function \(\tilde{m}_{\lambda }\) can be expressed in terms of power sums
without computing the transition matrices
According to (7) and Corollary 3, we obtain Algorithm 2 for computing the coefficient \({T}_{\lambda \beta }\) of the power sums \(p_{\beta }\).
In Algorithm 2, \(\lambda\) and \(\beta\) are two integer partitions of k such that
The recursive function Tlb(k) is presented in a form that allows fast identification of the correlation between Corollary 3 and the operations executed with the arrays \((t_1,t_2,\ldots ,t_k)\) and \((v_1,v_2,\ldots ,v_k)\). Thus, the lines 2–9 are useful to determine whether \(\beta =\lambda\) or \(\beta \prec \lambda\). The value of j is selected in the lines 16–24 such that j is the largest positive integer with
This selection of j allows us to reduce the number of recursive calls from the lines 30 and 39.
The arrays \((t_1,t_2,\ldots ,t_k)\) and \((v_1,v_2,\ldots ,v_k)\) are the global variables of the recursive function Tlb(k). These global variables are very important because help us save memory. The integer partitions \(\lambda\) and \(\lambda ^i\) with \(i\geqslant 0\) are alternatively stored in the same array \((t_1,t_2,\ldots ,t_k)\). The integer partition \(\lambda ^0\) is immediately derived from the integer partition \(\lambda\) in the line 28. Then \(\lambda\) is derived from \(\lambda ^0\) in the line 31. The integer partition \(\lambda ^i\) with \(i>0\) is derived from the integer partition \(\lambda\) in the lines 36–38. Then \(\lambda\) is derived from \(\lambda ^i\) in the lines 40–42. The integer partitions \(\beta\) and \(\beta ^0\) are alternatively stored in the same array \((v_1,v_2,\ldots ,v_k)\). The integer partition \(\beta ^0\) is immediately derived from the integer partition \(\beta\) in the line 29. Then \(\beta\) is derived from \(\beta ^0\) in the line 32.
The function Tlb(k) can be integrated into any algorithm for generating integer partitions to get the expression of the augmented monomial \(\tilde{m}_{\lambda }\) in terms of power sums.
Concluding remarks
An iterative algorithm for computing transition matrix expanding the augmented monomial symmetric functions in terms of the power sums symmetric functions has been derived in this paper. It is clear that the efficiency of this algorithm is directly influenced by the efficiency of the algorithm used for generating integer partitions in reverse lexicographic order. To express a specific augmented monomials in terms of power sums, we need a single line of the transition matrix. In this case, the computation of all transition matrix elements is not justified. Thus, a recursive function that computes the value of a single element of the transition matrix has been derived. Clearly, behind these algorithms is Theorem 1.
A recursive algorithm that requires algebraic symbol manipulation for expressing the augmented monomial \(\tilde{m}_{\lambda }\) in terms of power sums can be easily derived from Theorem 1. For instance, in Maple this algorithm can be written as
and the command
generates the following expression
Such a recursive algorithm is very simple but its effectiveness can not be called into question because of the large number of recursive calls (for the augmented monomial \(\tilde{m}_{\lambda }\) the number of recursive calls is the factorial of \(l(\lambda )1\)).
Unfortunately, Theorem 2 is more difficult to exploit in order to give similar results. However, a special case can be considered.
Corollary 4
Let \(\lambda =[\lambda _1,\lambda _2,\ldots ,\lambda _r]\) be an integer partition of k such that
for all \(i<r\).

1.
The number of integer partition \(\beta\) with \(\lambda \preceq \beta\) is greater than or equal to \(B_r\).

2.
The number of integer partition \(\beta\) with \(T_{\lambda \beta }=0\) is equal to
$$\begin{aligned} p(k)B_r, \end{aligned}$$where p(k) is the Euler partition function.

3.
For all \(v \in \mathcal {P}_r\) the following formula holds:
$$\begin{aligned} T_{\lambda ,s(v)}=\mu (v)\ , \end{aligned}$$where \(s(v)=[s_1,s_2,\ldots ,s_{\left v \right }]\) is an integer partition with
$$\begin{aligned} s_i=\sum _{j \in v_i}\lambda _j\ ,\quad i=1,\ldots ,\left v \right . \end{aligned}$$
References
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Garvan F (2001) The Maple Book. Chapman & Hall/CRC, Boca Raton
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Merca, M. Augmented monomials in terms of power sums. SpringerPlus 4, 724 (2015). https://doi.org/10.1186/s4006401515065
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Keywords
 Symmetric functions
 Integer partitions
Mathematics Subject Classification
 05E05
 11P81