R: Multinomial Formula in R

Question

I am working with the R programming language.

I am trying to write an R function that can fully expand a mathematical expression (in symbolic form) of the form: (a+b+c+....)^n

In the above expression, n is an integer. I want the final answer to be in terms of a,b,c... (i.e. I am not looking to perform arithmetic calculations, but rather expand an expression symbolically.

As I understand, this corresponds to the following mathematical equation: https://en.wikipedia.org/wiki/Multinomial_theorem

I was able to get the following function to work that expands a binomial expression:

expandBinom <- function(a, b, n) {
    sapply(0:n, function(k) {
        choose(n, k) * a^(n-k) * b^k
    })
}

The above function calculates the coefficients of the terms being expanded. For example, I can verify that (a+b)^2 = a^2 + 2*ab + b^2:

> expandBinom(1,1,2)
[1] 1 2 1

But I do not know how to generalize this above function to the multinomial case.

Is it possible to write a function like this for multinomial expressions - and actually have the expression appear in the form of (a^2 + 2ab^2 + c..) instead of just (1 2 1)

Thanks!

References:

• https://en.wikipedia.org/wiki/Pascal%27s_triangle

Answer 1

For symbolic computations

Probably you can try Ryacas package

library(Ryacas)

f <- function(n, ...) {
    as_r(yac_str(sprintf("Simplify(Expand((%s)^%s))", paste0(c(...), collapse = "+"), n)))
}

and you will obtain, for example

> f(5, "a", "b", "c")
expression(a^5 + 5 * a^4 * b + 5 * a^4 * c + 10 * a^3 * b^2 + 
    20 * a^3 * b * c + 10 * a^3 * c^2 + 10 * a^2 * b^3 + 30 * 
    a^2 * b^2 * c + 30 * a^2 * b * c^2 + 10 * a^2 * c^3 + 5 *
    a * b^4 + 20 * a * b^3 * c + 30 * a * b^2 * c^2 + 20 * a *
    b * c^3 + 5 * a * c^4 + b^5 + 5 * b^4 * c + 10 * b^3 * c^2 +
    10 * b^2 * c^3 + 5 * b * c^4 + c^5)

---

For numerical computations (coefficients only)

If you only care about the numerical coefficients of polynomial (e.g., a polynomial of x), you can try convolve + Reduce

h <- function(n, coeffs) {
    Reduce(\(x, y) convolve(x, rev(y), type = "open"), rep(list(coeffs), n))
}

or its recursion variant

h2 <- function(n, coeffs) {
    if (n == 1) {
        return(coeffs)
    }
    convolve(Recall(n - 1, coeffs), rev(coeffs), type = "open")
}

and you will see

# (1+x)^2 = 1 + 2x + x^2
> h(2, c(1, 1)) 
[1] 1 2 1

# (1+2x)^3 = 1 + 6x + 12x^2 + 8x^3
> h(3, c(1, 2)) 
[1]  1  6 12  8

# (1 + 4x + 5x^2) = 1 + 16x + 116x^2 + 496x^3 + 1366x^4 + 2480x^5 + 2900x^6 + 2000x^7 + 625x^8
> h(4, c(1, 4, 5)) 
[1]    1   16  116  496 1366 2480 2900 2000  625

Answer 2

You can use giacR (not on CRAN yet):

library(giacR)
giac <- Giac$new()
giac$execute(("expand((a + b + c)^5)"))
# "a^5+b^5+c^5+5*a*b^4+5*a*c^4+5*b*c^4+10*a^2*b^3+10*a^2*c^3+10*a^3*b^2+10*a^3*c^2+5*a^4*b+5*a^4*c+10*b^2*c^3+10*b^3*c^2+5*b^4*c+20*a*b*c^3+30*a*b^2*c^2+20*a*b^3*c+30*a^2*b*c^2+30*a^2*b^2*c+20*a^3*b*c"

Answer 3

Using the mpoly package:

mpoly::mp("(a + b + c)^5")
#> a^5  +  5 a^4 b  +  5 a^4 c  +  10 a^3 b^2  +  20 a^3 b c  +  10 a^3 c^2  +  10 a^2 b^3  +  30 a^2 b^2 c  +  30 a^2 b c^2  +  10 a^2 c^3  +  5 a b^4  +  20 a b^3 c  +  30 a b^2 c^2  +  20 a b c^3  +  5 a c^4  +  b^5  +  5 b^4 c  +  10 b^3 c^2  +  10 b^2 c^3  +  5 b c^4  +  c^5

Or, for the exponents and coefficients:

library(partitions)

f <- function(vars, pow) {
  m <- as.matrix(compositions(pow, length(vars)))
  rownames(m) <- vars
  
  list(
    exponents = m,
    coeffs = exp(lgamma(pow + 1) - colSums(array(lgamma(1:(pow + 1))[m + 1L], dim(m))))
  )
}

f(letters[1:3], 5)
#> $exponents
#>   [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21]
#> a    5    4    3    2    1    0    4    3    2     1     0     3     2     1     0     2     1     0     1     0     0
#> b    0    1    2    3    4    5    0    1    2     3     4     0     1     2     3     0     1     2     0     1     0
#> c    0    0    0    0    0    0    1    1    1     1     1     2     2     2     2     3     3     3     4     4     5
#> 
#> $coeffs
#>  [1]  1  5 10 10  5  1  5 20 30 20  5 10 30 30 10 10 20 10  5  5  1

Retrieving just the exponents and coefficients is much faster.

library(mpoly)

system.time(mp("(a + b + c + d + e)^20"))
#>    user  system elapsed 
#>   11.78    0.30   12.07
system.time(f(letters[1:5], 20))
#>    user  system elapsed 
#>    0.02    0.00    0.00

Answer 4

If you don't want to use external packages but base R only, you can define a custom function like below (but it won't scale well when you have high order n or many terms in ..., since expand.grid + apply are applied)

g <- function(n, ...) {
    comb <- expand.grid(rep(list(c(...)), n))
    d <- table(
        apply(comb, 1, \(x) {
            v <- sort(table(x))
            gsub("\\^1((?=\\D)|$)",
                "",
                paste0(paste(names(v), v, sep = "^"), collapse = "*"),
                perl = TRUE
            )
        })
    )
    sort(d)
}

and you will obtain, for example

> g(5, "x", "y")

    x^5     y^5   x*y^4   y*x^4 x^2*y^3 y^2*x^3
      1       1       5       5      10      10

> g(3, "a", "b", "c")

  a^3   b^3   c^3 a*b^2 a*c^2 b*a^2 b*c^2 c*a^2 c*b^2 a*b*c 
    1     1     1     3     3     3     3     3     3     6

The code above just presents the terms in a table, and you are definitely free to take additional actions to produce a "sum" expression.

Answer 5

As mentioned in one of the answers by @ThomasIsCoding, it would become TERRIBLY slow when we have higher order of n or many terms in ... (due to expand.grid and apply).

As always (due to my preference, I think it would be super fun if we don't load any external packages to make it), if you want to stay with base R but boost the performance significantly, I think the following base R function might help a lot.

---

Basic Idea

Assuming what we would like to expand is in the format (x1 + x2 + x3 + ... + xm)^n, where m is any positive integer, and n is a positive integer as the exponent, we can:

1. Create a helper function to produces all possible exponent combinations of (k1, k2, ..., km) for terms x1^k1 * x2^k2 * ... * xm^km, where the helper function can be defined in a recursion way.

2. On top of obtained tuple (k1, k2, ..., km), we should generate the corresponding multinomial coefficients, i.e., n!/(k1! * k2! * ... * km!), and then multiply them with the term x1^k1 * x2^k2 * ... * xm^km

---

Code

f <- function(m, n = m) {
    helper <- function(m, n) {
        if (m == 1) {
            return(n)
        }
        unlist(
            lapply(
                0:n,
                \(i) Map(c, i, helper(m - 1, max(n - i, 0)))
            ),
            recursive = FALSE
        )
    }

    sapply(
        helper(m, n),
        \(x) {
            term <- gsub(
                "\\^1((?=\\D)|$)",
                "",
                paste0(
                    paste0("x", seq_along(x), "^", x)[x > 0],
                    collapse = "*"
                ),
                perl = TRUE
            )
            coeff <- factorial(n) / prod(factorial(x))
            ifelse(coeff == 1, term, paste0(coeff, "*", term))
        }
    )
}

---

Output Examples

> f(2)
[1] "x2^2"    "2*x1*x2" "x1^2"   

> f(3, 2)
[1] "x3^2"    "2*x2*x3" "x2^2"    "2*x1*x3" "2*x1*x2" "x1^2"

> f(2, 4)
[1] "x2^4"        "4*x1*x2^3"   "6*x1^2*x2^2" "4*x1^3*x2"   "x1^4"

> f(4, 2)
 [1] "x4^2"    "2*x3*x4" "x3^2"    "2*x2*x4" "2*x2*x3" "x2^2"    "2*x1*x4"
 [8] "2*x1*x3" "2*x1*x2" "x1^2"

and a crazy one

> f(7, 5)
  [1] "x7^5"               "5*x6*x7^4"          "10*x6^2*x7^3"      
  [4] "10*x6^3*x7^2"       "5*x6^4*x7"          "x6^5"
  [7] "5*x5*x7^4"          "20*x5*x6*x7^3"      "30*x5*x6^2*x7^2"
 [10] "20*x5*x6^3*x7"      "5*x5*x6^4"          "10*x5^2*x7^3"
 [13] "30*x5^2*x6*x7^2"    "30*x5^2*x6^2*x7"    "10*x5^2*x6^3"
 [16] "10*x5^3*x7^2"       "20*x5^3*x6*x7"      "10*x5^3*x6^2"
 [19] "5*x5^4*x7"          "5*x5^4*x6"          "x5^5"
 [22] "5*x4*x7^4"          "20*x4*x6*x7^3"      "30*x4*x6^2*x7^2"
 [25] "20*x4*x6^3*x7"      "5*x4*x6^4"          "20*x4*x5*x7^3"
 [28] "60*x4*x5*x6*x7^2"   "60*x4*x5*x6^2*x7"   "20*x4*x5*x6^3"
 [31] "30*x4*x5^2*x7^2"    "60*x4*x5^2*x6*x7"   "30*x4*x5^2*x6^2"
 [34] "20*x4*x5^3*x7"      "20*x4*x5^3*x6"      "5*x4*x5^4"
 [37] "10*x4^2*x7^3"       "30*x4^2*x6*x7^2"    "30*x4^2*x6^2*x7"
 [40] "10*x4^2*x6^3"       "30*x4^2*x5*x7^2"    "60*x4^2*x5*x6*x7"
 [43] "30*x4^2*x5*x6^2"    "30*x4^2*x5^2*x7"    "30*x4^2*x5^2*x6"
 [46] "10*x4^2*x5^3"       "10*x4^3*x7^2"       "20*x4^3*x6*x7"
 [49] "10*x4^3*x6^2"       "20*x4^3*x5*x7"      "20*x4^3*x5*x6"
 [52] "10*x4^3*x5^2"       "5*x4^4*x7"          "5*x4^4*x6"
 [55] "5*x4^4*x5"          "x4^5"               "5*x3*x7^4"
 [58] "20*x3*x6*x7^3"      "30*x3*x6^2*x7^2"    "20*x3*x6^3*x7"
 [61] "5*x3*x6^4"          "20*x3*x5*x7^3"      "60*x3*x5*x6*x7^2"
 [64] "60*x3*x5*x6^2*x7"   "20*x3*x5*x6^3"      "30*x3*x5^2*x7^2"
 [67] "60*x3*x5^2*x6*x7"   "30*x3*x5^2*x6^2"    "20*x3*x5^3*x7"
 [70] "20*x3*x5^3*x6"      "5*x3*x5^4"          "20*x3*x4*x7^3"
 [73] "60*x3*x4*x6*x7^2"   "60*x3*x4*x6^2*x7"   "20*x3*x4*x6^3"
 [76] "60*x3*x4*x5*x7^2"   "120*x3*x4*x5*x6*x7" "60*x3*x4*x5*x6^2"
 [79] "60*x3*x4*x5^2*x7"   "60*x3*x4*x5^2*x6"   "20*x3*x4*x5^3"
 [82] "30*x3*x4^2*x7^2"    "60*x3*x4^2*x6*x7"   "30*x3*x4^2*x6^2"
 [85] "60*x3*x4^2*x5*x7"   "60*x3*x4^2*x5*x6"   "30*x3*x4^2*x5^2"   
 [88] "20*x3*x4^3*x7"      "20*x3*x4^3*x6"      "20*x3*x4^3*x5"
 [91] "5*x3*x4^4"          "10*x3^2*x7^3"       "30*x3^2*x6*x7^2"
 [94] "30*x3^2*x6^2*x7"    "10*x3^2*x6^3"       "30*x3^2*x5*x7^2"
 [97] "60*x3^2*x5*x6*x7"   "30*x3^2*x5*x6^2"    "30*x3^2*x5^2*x7"
[100] "30*x3^2*x5^2*x6"    "10*x3^2*x5^3"       "30*x3^2*x4*x7^2"
[103] "60*x3^2*x4*x6*x7"   "30*x3^2*x4*x6^2"    "60*x3^2*x4*x5*x7"
[106] "60*x3^2*x4*x5*x6"   "30*x3^2*x4*x5^2"    "30*x3^2*x4^2*x7"
[109] "30*x3^2*x4^2*x6"    "30*x3^2*x4^2*x5"    "10*x3^2*x4^3"
[112] "10*x3^3*x7^2"       "20*x3^3*x6*x7"      "10*x3^3*x6^2"
[115] "20*x3^3*x5*x7"      "20*x3^3*x5*x6"      "10*x3^3*x5^2"
[118] "20*x3^3*x4*x7"      "20*x3^3*x4*x6"      "20*x3^3*x4*x5"
[121] "10*x3^3*x4^2"       "5*x3^4*x7"          "5*x3^4*x6"
[124] "5*x3^4*x5"          "5*x3^4*x4"          "x3^5"
[127] "5*x2*x7^4"          "20*x2*x6*x7^3"      "30*x2*x6^2*x7^2"
[130] "20*x2*x6^3*x7"      "5*x2*x6^4"          "20*x2*x5*x7^3"
[133] "60*x2*x5*x6*x7^2"   "60*x2*x5*x6^2*x7"   "20*x2*x5*x6^3"
[136] "30*x2*x5^2*x7^2"    "60*x2*x5^2*x6*x7"   "30*x2*x5^2*x6^2"
[139] "20*x2*x5^3*x7"      "20*x2*x5^3*x6"      "5*x2*x5^4"
[142] "20*x2*x4*x7^3"      "60*x2*x4*x6*x7^2"   "60*x2*x4*x6^2*x7"
[145] "20*x2*x4*x6^3"      "60*x2*x4*x5*x7^2"   "120*x2*x4*x5*x6*x7"
[148] "60*x2*x4*x5*x6^2"   "60*x2*x4*x5^2*x7"   "60*x2*x4*x5^2*x6"
[151] "20*x2*x4*x5^3"      "30*x2*x4^2*x7^2"    "60*x2*x4^2*x6*x7"
[154] "30*x2*x4^2*x6^2"    "60*x2*x4^2*x5*x7"   "60*x2*x4^2*x5*x6"
[157] "30*x2*x4^2*x5^2"    "20*x2*x4^3*x7"      "20*x2*x4^3*x6"
[160] "20*x2*x4^3*x5"      "5*x2*x4^4"          "20*x2*x3*x7^3"
[163] "60*x2*x3*x6*x7^2"   "60*x2*x3*x6^2*x7"   "20*x2*x3*x6^3"
[166] "60*x2*x3*x5*x7^2"   "120*x2*x3*x5*x6*x7" "60*x2*x3*x5*x6^2"
[169] "60*x2*x3*x5^2*x7"   "60*x2*x3*x5^2*x6"   "20*x2*x3*x5^3"
[172] "60*x2*x3*x4*x7^2"   "120*x2*x3*x4*x6*x7" "60*x2*x3*x4*x6^2"
[175] "120*x2*x3*x4*x5*x7" "120*x2*x3*x4*x5*x6" "60*x2*x3*x4*x5^2"
[178] "60*x2*x3*x4^2*x7"   "60*x2*x3*x4^2*x6"   "60*x2*x3*x4^2*x5"
[181] "20*x2*x3*x4^3"      "30*x2*x3^2*x7^2"    "60*x2*x3^2*x6*x7"
[184] "30*x2*x3^2*x6^2"    "60*x2*x3^2*x5*x7"   "60*x2*x3^2*x5*x6"
[187] "30*x2*x3^2*x5^2"    "60*x2*x3^2*x4*x7"   "60*x2*x3^2*x4*x6"
[190] "60*x2*x3^2*x4*x5"   "30*x2*x3^2*x4^2"    "20*x2*x3^3*x7"
[193] "20*x2*x3^3*x6"      "20*x2*x3^3*x5"      "20*x2*x3^3*x4"
[196] "5*x2*x3^4"          "10*x2^2*x7^3"       "30*x2^2*x6*x7^2"
[199] "30*x2^2*x6^2*x7"    "10*x2^2*x6^3"       "30*x2^2*x5*x7^2"
[202] "60*x2^2*x5*x6*x7"   "30*x2^2*x5*x6^2"    "30*x2^2*x5^2*x7"   
[205] "30*x2^2*x5^2*x6"    "10*x2^2*x5^3"       "30*x2^2*x4*x7^2"
[208] "60*x2^2*x4*x6*x7"   "30*x2^2*x4*x6^2"    "60*x2^2*x4*x5*x7"
[211] "60*x2^2*x4*x5*x6"   "30*x2^2*x4*x5^2"    "30*x2^2*x4^2*x7"
[214] "30*x2^2*x4^2*x6"    "30*x2^2*x4^2*x5"    "10*x2^2*x4^3"
[217] "30*x2^2*x3*x7^2"    "60*x2^2*x3*x6*x7"   "30*x2^2*x3*x6^2"
[220] "60*x2^2*x3*x5*x7"   "60*x2^2*x3*x5*x6"   "30*x2^2*x3*x5^2"
[223] "60*x2^2*x3*x4*x7"   "60*x2^2*x3*x4*x6"   "60*x2^2*x3*x4*x5"
[226] "30*x2^2*x3*x4^2"    "30*x2^2*x3^2*x7"    "30*x2^2*x3^2*x6"
[229] "30*x2^2*x3^2*x5"    "30*x2^2*x3^2*x4"    "10*x2^2*x3^3"
[232] "10*x2^3*x7^2"       "20*x2^3*x6*x7"      "10*x2^3*x6^2"
[235] "20*x2^3*x5*x7"      "20*x2^3*x5*x6"      "10*x2^3*x5^2"
[238] "20*x2^3*x4*x7"      "20*x2^3*x4*x6"      "20*x2^3*x4*x5"
[241] "10*x2^3*x4^2"       "20*x2^3*x3*x7"      "20*x2^3*x3*x6"
[244] "20*x2^3*x3*x5"      "20*x2^3*x3*x4"      "10*x2^3*x3^2"
[247] "5*x2^4*x7"          "5*x2^4*x6"          "5*x2^4*x5"
[250] "5*x2^4*x4"          "5*x2^4*x3"          "x2^5"
[253] "5*x1*x7^4"          "20*x1*x6*x7^3"      "30*x1*x6^2*x7^2"
[256] "20*x1*x6^3*x7"      "5*x1*x6^4"          "20*x1*x5*x7^3"
[259] "60*x1*x5*x6*x7^2"   "60*x1*x5*x6^2*x7"   "20*x1*x5*x6^3"
[262] "30*x1*x5^2*x7^2"    "60*x1*x5^2*x6*x7"   "30*x1*x5^2*x6^2"
[265] "20*x1*x5^3*x7"      "20*x1*x5^3*x6"      "5*x1*x5^4"
[268] "20*x1*x4*x7^3"      "60*x1*x4*x6*x7^2"   "60*x1*x4*x6^2*x7"
[271] "20*x1*x4*x6^3"      "60*x1*x4*x5*x7^2"   "120*x1*x4*x5*x6*x7"
[274] "60*x1*x4*x5*x6^2"   "60*x1*x4*x5^2*x7"   "60*x1*x4*x5^2*x6"
[277] "20*x1*x4*x5^3"      "30*x1*x4^2*x7^2"    "60*x1*x4^2*x6*x7"
[280] "30*x1*x4^2*x6^2"    "60*x1*x4^2*x5*x7"   "60*x1*x4^2*x5*x6"
[283] "30*x1*x4^2*x5^2"    "20*x1*x4^3*x7"      "20*x1*x4^3*x6"
[286] "20*x1*x4^3*x5"      "5*x1*x4^4"          "20*x1*x3*x7^3"
[289] "60*x1*x3*x6*x7^2"   "60*x1*x3*x6^2*x7"   "20*x1*x3*x6^3"
[292] "60*x1*x3*x5*x7^2"   "120*x1*x3*x5*x6*x7" "60*x1*x3*x5*x6^2"
[295] "60*x1*x3*x5^2*x7"   "60*x1*x3*x5^2*x6"   "20*x1*x3*x5^3"
[298] "60*x1*x3*x4*x7^2"   "120*x1*x3*x4*x6*x7" "60*x1*x3*x4*x6^2"
[301] "120*x1*x3*x4*x5*x7" "120*x1*x3*x4*x5*x6" "60*x1*x3*x4*x5^2"
[304] "60*x1*x3*x4^2*x7"   "60*x1*x3*x4^2*x6"   "60*x1*x3*x4^2*x5"
[307] "20*x1*x3*x4^3"      "30*x1*x3^2*x7^2"    "60*x1*x3^2*x6*x7"
[310] "30*x1*x3^2*x6^2"    "60*x1*x3^2*x5*x7"   "60*x1*x3^2*x5*x6"
[313] "30*x1*x3^2*x5^2"    "60*x1*x3^2*x4*x7"   "60*x1*x3^2*x4*x6"
[316] "60*x1*x3^2*x4*x5"   "30*x1*x3^2*x4^2"    "20*x1*x3^3*x7"
[319] "20*x1*x3^3*x6"      "20*x1*x3^3*x5"      "20*x1*x3^3*x4"
[322] "5*x1*x3^4"          "20*x1*x2*x7^3"      "60*x1*x2*x6*x7^2"
[325] "60*x1*x2*x6^2*x7"   "20*x1*x2*x6^3"      "60*x1*x2*x5*x7^2"
[328] "120*x1*x2*x5*x6*x7" "60*x1*x2*x5*x6^2"   "60*x1*x2*x5^2*x7"
[331] "60*x1*x2*x5^2*x6"   "20*x1*x2*x5^3"      "60*x1*x2*x4*x7^2"
[334] "120*x1*x2*x4*x6*x7" "60*x1*x2*x4*x6^2"   "120*x1*x2*x4*x5*x7"
[337] "120*x1*x2*x4*x5*x6" "60*x1*x2*x4*x5^2"   "60*x1*x2*x4^2*x7"
[340] "60*x1*x2*x4^2*x6"   "60*x1*x2*x4^2*x5"   "20*x1*x2*x4^3"     
[343] "60*x1*x2*x3*x7^2"   "120*x1*x2*x3*x6*x7" "60*x1*x2*x3*x6^2"
[346] "120*x1*x2*x3*x5*x7" "120*x1*x2*x3*x5*x6" "60*x1*x2*x3*x5^2"
[349] "120*x1*x2*x3*x4*x7" "120*x1*x2*x3*x4*x6" "120*x1*x2*x3*x4*x5"
[352] "60*x1*x2*x3*x4^2"   "60*x1*x2*x3^2*x7"   "60*x1*x2*x3^2*x6"
[355] "60*x1*x2*x3^2*x5"   "60*x1*x2*x3^2*x4"   "20*x1*x2*x3^3"
[358] "30*x1*x2^2*x7^2"    "60*x1*x2^2*x6*x7"   "30*x1*x2^2*x6^2"
[361] "60*x1*x2^2*x5*x7"   "60*x1*x2^2*x5*x6"   "30*x1*x2^2*x5^2"
[364] "60*x1*x2^2*x4*x7"   "60*x1*x2^2*x4*x6"   "60*x1*x2^2*x4*x5"
[367] "30*x1*x2^2*x4^2"    "60*x1*x2^2*x3*x7"   "60*x1*x2^2*x3*x6"
[370] "60*x1*x2^2*x3*x5"   "60*x1*x2^2*x3*x4"   "30*x1*x2^2*x3^2"
[373] "20*x1*x2^3*x7"      "20*x1*x2^3*x6"      "20*x1*x2^3*x5"
[376] "20*x1*x2^3*x4"      "20*x1*x2^3*x3"      "5*x1*x2^4"
[379] "10*x1^2*x7^3"       "30*x1^2*x6*x7^2"    "30*x1^2*x6^2*x7"
[382] "10*x1^2*x6^3"       "30*x1^2*x5*x7^2"    "60*x1^2*x5*x6*x7"
[385] "30*x1^2*x5*x6^2"    "30*x1^2*x5^2*x7"    "30*x1^2*x5^2*x6"
[388] "10*x1^2*x5^3"       "30*x1^2*x4*x7^2"    "60*x1^2*x4*x6*x7"
[391] "30*x1^2*x4*x6^2"    "60*x1^2*x4*x5*x7"   "60*x1^2*x4*x5*x6"
[394] "30*x1^2*x4*x5^2"    "30*x1^2*x4^2*x7"    "30*x1^2*x4^2*x6"
[397] "30*x1^2*x4^2*x5"    "10*x1^2*x4^3"       "30*x1^2*x3*x7^2"
[400] "60*x1^2*x3*x6*x7"   "30*x1^2*x3*x6^2"    "60*x1^2*x3*x5*x7"
[403] "60*x1^2*x3*x5*x6"   "30*x1^2*x3*x5^2"    "60*x1^2*x3*x4*x7"
[406] "60*x1^2*x3*x4*x6"   "60*x1^2*x3*x4*x5"   "30*x1^2*x3*x4^2"
[409] "30*x1^2*x3^2*x7"    "30*x1^2*x3^2*x6"    "30*x1^2*x3^2*x5"
[412] "30*x1^2*x3^2*x4"    "10*x1^2*x3^3"       "30*x1^2*x2*x7^2"
[415] "60*x1^2*x2*x6*x7"   "30*x1^2*x2*x6^2"    "60*x1^2*x2*x5*x7"
[418] "60*x1^2*x2*x5*x6"   "30*x1^2*x2*x5^2"    "60*x1^2*x2*x4*x7"
[421] "60*x1^2*x2*x4*x6"   "60*x1^2*x2*x4*x5"   "30*x1^2*x2*x4^2"
[424] "60*x1^2*x2*x3*x7"   "60*x1^2*x2*x3*x6"   "60*x1^2*x2*x3*x5"
[427] "60*x1^2*x2*x3*x4"   "30*x1^2*x2*x3^2"    "30*x1^2*x2^2*x7"
[430] "30*x1^2*x2^2*x6"    "30*x1^2*x2^2*x5"    "30*x1^2*x2^2*x4"
[433] "30*x1^2*x2^2*x3"    "10*x1^2*x2^3"       "10*x1^3*x7^2"
[436] "20*x1^3*x6*x7"      "10*x1^3*x6^2"       "20*x1^3*x5*x7"
[439] "20*x1^3*x5*x6"      "10*x1^3*x5^2"       "20*x1^3*x4*x7"
[442] "20*x1^3*x4*x6"      "20*x1^3*x4*x5"      "10*x1^3*x4^2"
[445] "20*x1^3*x3*x7"      "20*x1^3*x3*x6"      "20*x1^3*x3*x5"
[448] "20*x1^3*x3*x4"      "10*x1^3*x3^2"       "20*x1^3*x2*x7"
[451] "20*x1^3*x2*x6"      "20*x1^3*x2*x5"      "20*x1^3*x2*x4"
[454] "20*x1^3*x2*x3"      "10*x1^3*x2^2"       "5*x1^4*x7"
[457] "5*x1^4*x6"          "5*x1^4*x5"          "5*x1^4*x4"
[460] "5*x1^4*x3"          "5*x1^4*x2"          "x1^5"