Oivalsin symbolien järjestyksen.

Oivalsin is the 1st person singular past tense of the verb oivaltaa.

• oivaltaa = to realize, to figure out, to grasp
• oivalsin = I realized / I figured out

The ending -in is the typical past-tense ending for I in many Finnish verbs.

Finnish often leaves out the subject pronoun when it is clear from the verb ending.

So:

• oivalsin already means I realized
• minä oivalsin is also possible, but minä is usually unnecessary unless you want emphasis or contrast

That is why the sentence can simply start with Oivalsin.

Symbolien is the plural genitive of symboli.

• symboli = symbol
• symbolit = symbols
• symbolien = of the symbols

In this sentence, it modifies järjestyksen, so together:

• symbolien järjestys = the order of the symbols

This is how Finnish expresses relationships like the order of X, the color of X, the name of X, and so on: the first noun often goes into the genitive.

Here järjestyksen is the object of the verb, and it is a total object.

• järjestys = order
• järjestyksen = the order / the whole order as a completed object

With a normal affirmative sentence in the past tense, a singular total object usually looks like the genitive form.

So in:

• Oivalsin symbolien järjestyksen

the speaker realized the whole thing, not just part of it.

This is why you get järjestyksen, not the partitive järjestystä.

This is a very common question, because in this kind of sentence the form looks like the genitive.

Traditionally, in a sentence like this, järjestyksen is functioning as the total object, and for singular nouns that object often has the same form as the genitive.

So:

• form: genitive-looking
• function: object

Many learners remember it like this:

• genitive form used as a total object

That is a practical way to understand it.

Because Finnish does not usually put two basic noun forms next to each other to mean the order of the symbols.

Instead, Finnish uses:

• symbolien järjestys
  • literally: symbols' order / order of the symbols

So the pattern is:

• X:n / X:ien + noun

Examples:

• talon ovi = the door of the house
• kirjojen nimet = the names of the books
• symbolien järjestys = the order of the symbols

Yes. Finnish word order is fairly flexible, though not random.

The neutral order here is:

• Oivalsin symbolien järjestyksen.

But you could also say:

• Symbolien järjestyksen oivalsin.

This gives more emphasis to symbolien järjestyksen. It can sound marked, contrastive, or literary depending on context.

So the original sentence is the most neutral and natural choice in ordinary usage.

Finnish has no articles, so it does not have direct equivalents of a/an and the.

That means:

• symbolien järjestyksen can be understood as the order of the symbols
• the exact interpretation depends on context

English needs articles, but Finnish usually does not.

Oivaltaa usually suggests a moment of insight: suddenly realizing or figuring something out.

• oivaltaa = to realize, to have an insight, to figure out
• ymmärtää = to understand

So Oivalsin symbolien järjestyksen sounds more like:

• I figured out the order of the symbols
• I suddenly realized how the symbols were ordered

If you used ymmärsin, it would sound more like simply understanding it, without the same aha! feeling.

The dictionary forms are:

• oivaltaa → oivalsin
• symboli → symbolien
• järjestys → järjestyksen

A learner may notice that the stems change a little:

• järjestys becomes järjestykse- in some forms, which is why you get järjestyksen
• symboli becomes symbolie- in the plural genitive, giving symbolien

These changes are normal in Finnish noun inflection and are something you gradually get used to by seeing many examples.