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# incOrbit -- the increasing map orbit of an element

- Usage:
    `O = incOrbit(f,n)`
    `O = incOrbit(F,n)`
- Inputs:
  - f, a ring element, an element of a ring created by buildERing
  - F, a list, a list of ring elements
  - n, an integer, the index bound
- Optional inputs:
  - Symmetrize => ..., default value false
- Outputs:
  - O, a list, a list of all elements in the orbit of f or F

## Description

If F is an equivariant Gröbner basis for invariant ideal I with respect to a width order then this method produces a traditional Gröbner basis for the nth truncation of I.

If the optional argument Symmetrize is set to true, then the full S_n orbit is produced.

    i1 : R = buildERing({symbol x}, {1}, QQ, 2);

    i2 : O = incOrbit(x_0^2, 4)

           2   2   2   2
    o2 = {x , x , x , x }
           0   1   2   3

    o2 : List

    i3 : P = incOrbit(x_0 + x_1^2, 3, Symmetrize=>true)

           2             2   2             2   2             2
    o3 = {x  + x , x  + x , x  + x , x  + x , x  + x , x  + x }
           1    0   1    0   2    0   2    0   2    1   2    1

    o3 : List

## Caveat

The output is not necessarily in the same ring as the input. The width bound of the ring of the output will always be n.

## Ways to use incOrbit:

- `incOrbit(List,ZZ)`
- `incOrbit(RingElement,ZZ)`

## For the programmer

The object incOrbit is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/EquivariantGB.m2:1383:0.