In addition to stefgosselin's answer that has some mistakes: Lets start with his array:

$input = array(1,2,3);

This contains:

array(3) {
    [0]=> int(1)
    [1]=> int(2)
    [2]=> int(3)
}

Then you do array_slice:

var_dump(array_slice($input, 1));

The function will return the values after the first element (thats what the second argument, the offset means). But notice the keys!

array(2) {
    [0]=> int(2)
    [1]=> int(3)
}

Keep in mind that keys aren't preserved, until you pass true for the fourth preserve_keys parameter. Also because there is another length parameter before this, you have to pass NULL if you want to return everything after the offset, but with the keys preserved.

var_dump(array_slice($input, 1, NULL, true));

That will return what stefgosselin (incorrectly) wrote initially.

array(2) {
    [1]=> int(2)
    [2]=> int(3)
}

Better make assosative array instead of typing $array_1, $array_2. And instead of making array with same values like array('1', '1', '1'), you can just keep the amount of value like array('1' => 3)

Copy$mix_array = array ('1','3','6','1','5','3','6','5','1','7','3','9');
$group_array = array();
foreach ($mix_array as $value) {
    if(isset($group_array[$value])) {
        $group_array[$value]++;
    }
    else {
        $group_array[$value] = 1;
    }
}

print_r($group_array);

Result

CopyArray
(
[1] => 3
[3] => 3
[6] => 2
[5] => 2
[7] => 1
[9] => 1
)

Or as Josh said simply use array_count_values()

You actually have an array that contains arrays which probably might be what is causing the issue for you. Though I do not see how you get the result you posted... It mighty be that you actually apply the slice function to each element of the output array. Then certainly you will get an empty array for each iteration. As to be expected, since each element you iterate over contains only a single element. So slicing from position 1 will result in an empty array each time...

Consider this simple example using a plain array:

<?php
$input = ["one", "two", "three", "four", "five", "six"];
$output = array_slice($input, 1, 2);
print_r($output);

The output is:

Array
(
    [0] => two
    [1] => three
)

So php's array_slice() function works just as expected...

Same with an array of arrays as you suggest in your post:

<?php
$input = [["one"], ["two"], ["three"], ["four"], ["five"], ["six"]];
$output = array_slice($input, 1, 2);
print_r($output);

The output of that is, as expected:

Array
(
    [0] => Array
        (
            [0] => two
        )

    [1] => Array
        (
            [0] => three
        )

)

Also considering your second comment below, that you have some words in a single string (which mighty be what you describe) I get a meaningful result myself:

<?php
$input = explode(' ', "one two three four five six");
$output = array_slice($input, 1, 2);
print_r($output);

The output, as expected, is:

Array
(
    [0] => two
    [1] => three
)