# Augmented monomials in terms of power sums

- Mircea Merca
^{1}Email author

**Received: **7 August 2015

**Accepted: **3 November 2015

**Published: **24 November 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.

### Keywords

Symmetric functions Integer partitions### Mathematics Subject Classification

05E05 11P81## Background

*n*can be written as a sum of one or more positive integers, i.e.,

*i*appears \(t_i\) times. If the order of integers \(\lambda _i\) is important, then the representation (1) is known as a composition. For

*n*, we use the notation \(\lambda \vdash n\). We denote by \(l(\lambda )\) the number of parts of \(\lambda\), i.e.,

*k*largest parts of \(\alpha\) is less than the sum of the

*k*largest parts of \(\beta\), i.e.,

*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.

*k*th power sum symmetric function \(p_k=p_k (x_1,x_2,\dots ,x_n)\), i.e.,

*k*. It is well-known that the set

*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

## 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 non-empty, 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*

*M*the set of terms in the expression \(p_{\lambda _k}\cdot \tilde{m}_{[\lambda _1,\lambda _2,\ldots ,\lambda _{k-1}]}\), 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 \(\tilde{m}_{[\lambda _1,\ldots ,\lambda _{i-1},\lambda _{i}+\lambda _{k},\lambda _{i+1},\ldots ,\lambda _{k-1}]}\), for \(i=1,2,\ldots ,k-1\). According to

*M*. Therefore, the theorem is proved. \(\square\)

###
*Example 1*

*k*by 2 in Theorem 1, we get

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 \(\lambda ^0 \vdash k-j\) 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*

*n*th Bell number, \(B_n\) (see Sloane 2012, A000110). The Möbius function of \(\mathcal {P}_n\) (Bender and Goldman 1975; Rota 1964), namely

###
**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*

*and*

*p*

*are functions of*

*n*

*variables*, \(n\geqslant l(\lambda )\).

###
*Proof*

*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

###
*Example 3*

## 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*

*and*

*p*

*are functions of*

*n*

*variables*, \(n\geqslant l(\lambda )\).

###
*Example 4*

*k*th 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

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 \(k-j\) and \(\lambda \prec \lambda ^i\) for all \(i>0\) with \(t_i>\delta _{ij}\). We remark that \(\beta ^0 \vdash k-j\) for \(v_j>0\).

###
*Example 5*

###
*Example 6*

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

*k*such that

*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

*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.

*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

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 towhere$$\begin{aligned} p(k)-B_r, \end{aligned}$$
*p*(*k*) is the Euler partition function. - 3.For all \(v \in \mathcal {P}_r\) the following formula holds:where \(s(v)=[s_1,s_2,\ldots ,s_{\left| v \right| }]\) is an integer partition with$$\begin{aligned} T_{\lambda ,s(v)}=\mu (v)\ , \end{aligned}$$$$\begin{aligned} s_i=\sum _{j \in v_i}\lambda _j\ ,\quad i=1,\ldots ,\left| v \right| . \end{aligned}$$

## Declarations

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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