Friday Nov 17 and Mon Nov 20

Inversions of a permutation = number of pairs (i < j) such that pi(i) > pi(j).
Neighborly transpositions = (i i+1), in cycle notation.
Theorem: pi can be written as a product of invs(pi) many neighborly transpositions.
Theorem: if pi is written as a product of k neighborly transpositions, then k is greater than or equal to invs(pi), and k is congruent to invs(pi) mod 2.
Definition. An even permutation is one with invs(pi) even, similar def for odd permutations.
Theorem: invs(pi) + invs(rho) is congruent to invs(pi o rho) mod 2.
Theorem: invs(any transposition) is odd.
Theorem: if pi is written as a product of k transpositions, then k is congruent to invs(pi) mod 2.
Theorem: the Sam Loyd 14-15 puzzle is insoluble.