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Permutation parity — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The permutation (1 3 4 2) drawn as 4 strings, crossing 3 times. A permutation drawn as strings running from a row of numbered pegs to another, with every place two strings cross marked, and the crossing count checked against the number of pairs that are out of order.

    The crossings that will not come out even

    Draw a rearrangement as strings from one row of pegs to another and count where they cross. The count depends on how the strings are drawn; whether it is odd or even does not, and that single bit is what makes determinants exist and a sliding puzzle unsolvable.

    part 1 · algebra
  2. The half of 24 permutations that commutators reach. A block of 24 squares, one per permutation, with the 12 generated by commutators shaded, beside bars counting the homomorphisms to each cyclic group.

    The only bit that survives

    A shuffle can be called even or odd, and the label behaves under composition. Ask whether some cleverer label — a number out of three, or out of four — could behave the same way, and the answer is that nothing else can — one bit is exactly what a permutation gives up.

    part 2 · algebra
  3. A 2×3 sliding puzzle: 360 arrangements of 720 can be reached. Two arrangements of a small sliding puzzle side by side, the solved one and the one with two tiles exchanged, with the count of positions reachable by sliding found by walking every move.

    The puzzle that is exactly half solvable

    A sliding puzzle sold with two tiles swapped is not a hard puzzle; it is an impossible one, and the proof is a quantity that no slide can change. The same argument, run three times at once, says that one arrangement of a scrambled cube in twelve is reachable.

    part 3 · algebra
  4. The character table of the permutations of 4 places, computed from traces. A table with one row per irreducible representation and one column per conjugacy class, giving the trace of the matrix each representation assigns, with the dimension column marked.

    When the label may be a matrix

    A permutation carries exactly one bit into any commutative target, and commutativity is the restriction doing all the work. Drop it — let the label be a matrix — and what survives is a short finite table, computed here from traces and checked for orthogonality over every pair of rows.

    part 4 · algebra
  5. Every reachable arrangement of the 3×3 sliding puzzle, by moves from solved. A bar chart of the 181440 reachable arrangements of the 3×3 sliding puzzle by the fewest moves that solve them, rising to a peak at 24 and falling to 2 at the maximum distance 31.

    Thirty-one moves from solved

    Parity settles which half of a sliding puzzle's arrangements can be reached and is silent about how far away any of them is. Searching every reachable arrangement of the three-by-three tray answers the second question exactly — two arrangements sit thirty-one moves out — and parity turns up again, this time as a law about distance.

    part 5 · algebra
  6. Which arrangements sliding tokens reach, on nine graphs, against Wilson's theorem. A table of 9 small graphs drawn as icons, each with the searched count of reachable token arrangements and the fraction of all arrangements it is: a six-cycle 5/120; K₂,₃ 12/24; the 2×3 tray 60/120; a house 24/24; a six-cycle with one chord 120/120; a wheel of six 120/120; θ, inner paths 2, 2, 2 2520/5040; θ₀, inner paths 1, 2, 2 120/720; θ, inner paths 1, 3, 3 20160/40320.

    Which graphs let the tokens go anywhere

    A sliding puzzle is a graph with a token on every vertex but one. Richard Wilson found in 1974 what every such puzzle can reach, and the answer has a surprise in it — the half the tray is stuck with is not a fact about permutations at all, but about the board being two-coloured — and one exception, a graph of seven vertices that reaches exactly 120 of 720.

    part 6 · algebra

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