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Two people who rank each other first are called soulmates. The profiles in this sequence are required to have n pairs of soulmates.

The profiles with n pairs of soulmates are counted by sequence A343698. The profiles such that the men's preference form a Latin square are counted by A343696. The profiles in this sequence are the intersection of profiles in A343696 and A343698.

The Gale-Shapley algorithm (both men-proposing and women-proposing) on the preference profiles described by this sequence ends in one round.

Two people who rank each other first are called soulmates. Thus, the profiles in this sequence have n pairs of soulmates.

The profiles with n pairs of soulmates are counted by sequence A343698. The profiles such that the men's preferences form a Latin square are counted by A343696. The profiles such that both men's and women's preferences form a Latin square are counted by A343697. The profiles in this sequence are the intersection of profiles in A343698 and A343697.

Both the men- and the women-proposing Gale-Shapley algorithm on the preference profiles described by this sequence end in one round.

The profiles in this sequence are the intersection of the profiles in A343696 and A343697. The Gale-Shapley algorithm on such a set of preference profiles ends in one round.

A pine Latin square is a not necessarily canonical composite Latin square of order N=2*K formed from specially arranged cyclic Latin squares of order K.

By construction, pine Latin square is determined one-to-one by the cyclic square used, so number of pine Latin squares of order N is equal to number of cyclic Latin squares of order N/2.

All pine Latin squares are horizontally symmetric column-inverse Latin squares.

All pine Latin squares for selected order N are isomorphic one to another as Latin squares, so they have same properties (number of transversals, intercalates, etc.).

Pine Latin squares have interesting properties, for example, maximum known number of intercalates (see A383368 and A092237) for some orders N (at least N in {2, 4, 6, 10, 18}).

Pine Latin squares do not exist for odd orders because they must be horizontally symmetric.

Hypothesis: number of transversals in pine Latin squares of all orders N=4k+2 is zero (verified for orders N<=18).

The even and odd subsequences are A036980, A036981.

Definition: Let G(*) be a semigroup. A finite sequence u_1, u_2,... u_n of elements from G is called unique-valued with value v and length n provided v = u_1 * u_2 *... * u_n and v != u_p(1) * u_p(2) *... * u_p(n) for any non-identity permutation p of the indices {1, 2,... n}.

In other words, the only way to obtain the unique value v from the elements u_1, u_2,... u_n is by multiplying them in that particular order; any other order always gives a value different from v.

When the length of the sequence is 2, the meaning of "unique-valued" is equivalent to "the two elements do not commute under *".

I proved that the maximal possible length of a unique-valued sequence in the monoid M_n(*) of all n X n matrices (with entries in some ring with 1) is exactly 2n-1 (that 2n-1 is an upper bound follows from the Amitsur-Levitzki theorem), providing a positive example that this limit is reached.

I also proved that the maximal possible length of unique-valued sequences in S_n is 2n-3 (using n-simplex and again using the Amitsur-Levitzki theorem), but didn't find examples that this limit is really reached. My computer said "yes" for n=2 to 5, but even 6 is too large to compute.

The numbers of unique-valued sequences in S_n of the maximal length 2n-3 form a sequence 1, 2, 12, 576, which seems to coincide with A002860, the number of Latin squares of order n.

A Latin square is pluperfect if in each possible dissection of the Latin square into n translates of congruent rectangular blocks, no block contains duplicate symbols. For any prime number p, a(p) = A002860(p) and a(p^2) = A107739(p).

The values are calculated recursively, based on the characterization by 2009. The number a(5) was found before (2001 and, independently, later works) by exhaustive computer-aided classification of the objects.