Combinatorics / CONCEPT

The five cases for a third-position minimum

Let p be a permutation of size at least five with minimum third, and form q by deleting that minimum and subtracting one from every remaining value. The permutation p is simple and in C if and only if q is a permutation in C reconstructing p by minimum insertion, and at least one of five cases holds: q is simple with minimum at zero-based position at least three; q is the second-entry inflation by 21 of a unique simple skeleton in C of size at least four whose minimum is not second; p is obtained by prepending two to a unique simple parent in C with minimum second while increasing its values other than one; p is obtained in the same way after second-entry inflation by 12 of a unique simple skeleton in C of size at least four with minimum second; or q belongs to D, ends in its minimum, and equals R at its size.