A solution to this math problem:

The number of unique permutations of pans possible in a Pocket Palette is:

# Calculating

Here is the JavaScript code that calculates the answer:
```const width = 7;
const height = 4;

// We represent the palette as a bitfield stored in a 32 bit integer.
// 0 bits represent empty space (filled by 1x1 mini pans at the end), and
// 1 bits represent spaces filled by other pans, or spaces off the edge of
// the palette. We are lucky that the palette is only 7 spaces wide, as
// this allows room for 1 space of padding between rows while still fitting
// in 32 bits.

if ((width + 1) * height > 32) throw 'palette does not fit in 32 bits';

const move = (pan, x, y) => pan << x + y * (width + 1);
// We calculate the number of permutations by brute force. At each position
// in the palette we try to place every pan type. If it fits, we recurse and
// try every pan in every subsequent position until we reach the end of the
// palette.
const permutations = (palette, start) => {
// The passed in palette counts as one valid permutation, when completed
// using 1x1 mini pans (if necessary).
let count = 1;
// Try each pan type.
for (let i = 0; i < pans.length; i++) {
const pan = pans[i];
const maxShift = maxShifts[i];
// Try putting this pan at every position in the palette starting
// here.
for (let shift = start; shift < maxShift; shift++)
// If the pan fits here, place it and recurse.
if (!(palette & pan << shift))
count += permutations(palette | pan << shift, shift + 1);
}
return count;
}

// We consider five types of pans (rotated pans count as different types).
// Standard are 2x1, double are 2x2, and mixing are 2x4. Mini pans (1x1)
// can always fill any leftover space and there is always only one way to
// place them so we don't need to count them.
const standardH = 1 | move(1, 1, 0);
const standardV = 1 | move(1, 0, 1);
const double = standardH | move(standardH, 0, 1);
const mixingH = double | move(double, 2, 0);
const mixingV = double | move(double, 0, 2);
const pans = [ standardH, standardV, double, mixingH, mixingV ];

// Calculate the maximum we can move each pan before it hits the end
// of the 32 bit bitfield.
let maxShifts = [];
for (let pan of pans)
maxShifts.push(31 - (Math.log2(pan) | 0));

let palette = 0;
for (let x = 0; x < width; x++)
for (let y = 0; y < height; y++)
palette |= move(1, x, y);