cp's OEIS Frontend

"Reduced" instances are counted, as in A351430 for order 4.

Reduced instances are fewer than all instances by a factor of n!(n-1)! due to participant-renaming isomorphism, analogous to reduced latin squares.

The counts are listed in reverse order from A351430 so that the known terms appear first. The first term gives the number of maximal instances (with 16 stable matchings, per A357269).

The total number of reduced instances for order 5 is 214990848000000000, given by A351409(5).

The corresponding sequence for order 3 is [91,957,2840], with 91 maximal instances.

a(0) = A368433(5) = A344669(5) / (5! * 4!).

Results were obtained using the MiniZinc constraint modeling language, by extending the "Model for the Stable Marriage Problem" given in the MiniZinc Handbook, Section 2.2.6.

The first two terms were obtained on a PC with 12GB RAM; the next three terms on a PC with 64GB RAM.

Reduced instances (A351409) are fewer than all instances by a factor of n!(n-1)! due to participant-renaming isomorphism, analogous to reduced latin squares.

For n in [1,2,4], a(n) = 1 showing uniqueness up to isomorphism.

In the Stable Marriage Problem, the men's and women's preference lists can be swapped without affecting the number of blocking pairs or stable matchings, because the definitions of blocking pairs and stable matchings are symmetrical with respect to gender. a(n) is the number of instances in a canonical form where the men's preferences are lexicographically less than or equal to the women's preferences.

The A185141(n) instances of order n can be arranged in a square table with rows and columns indexed respectively by all possible men's and women's preferences in lexical order. The main diagonal of the square would be instances with men's preferences equal to women's preferences. The upper triangular region above the diagonal would contain instances with men's preferences less than women's preferences. The number of rows and columns in the table is given by A036740. The number of elements in the upper triangular region of a square, including the diagonal, is given by A000217. So a(n) composes A000217 with A036740 (performing A036740 first).

This sequence is like A351409 and A343700 in that they all involve means of reducing the search space, applied either individually or in combination, when searching for instances that maximize the number of stable matchings.

Each participant ranks all participants other than themselves in strict order, giving (2n-1)! orderings for each of 2n participants.

a(n) is from Appendix C of Thurber's 2002 paper, using the maximum from each row. At the time of publication, the bounds were known to be exact up to n=4. A357269 shows that they are also exact for n=5. This sequence is not to be confused with A069156, also from Thurber's Appendix C, which uses only the first column, making for looser bounds for n > 11. a(6), a(8), a(10), a(12), and a(16) are also conjectured to be exact.

Improved lower bounds for n=7, 9, 11, 13, 15 are shown in linked Ong et al. (2025) file.

Finding a(n) (denoted f(n) in the literature) is a research problem posed by Knuth in 1976 and reiterated by Gusfield and Irving in 1989.

a(5)=16 was found by Eilers using a MiniZinc constraint satisfaction model, showing previous lower bound of Eilers reported by Thurber to be exact.

A357271 gives lower bounds for a(n) reported by Thurber, previously known to be exact up to n=4, which is not to be confused with A069156 which gives looser lower bounds for n > 11.

Thurber proved a(n) to be strictly increasing.

There are 176130 reduced instances of order 5 with 16 stable matchings, and 498320 reduced instances with 15 stable matchings, compared with A351430 for order 4, and A369597 for order 3.

The total number of reduced instances for order n is A351409(n), which is 214990848000000000 for order 5, so about one in 1.22*10^12 such instances are maximal.

The maximum number of stable matchings for order 5, where the men's (or women's, respectively) ranking table is a latin square, is 14, with 300 such reduced instances, making order 5 the first order not containing any maximal instances where the men's ranking table is a Latin square.

a(n) is the number of women's ranking tables in the stable marriage problem that can be paired with a men's ranking table having no two men with the same first choice, without forming any mutual first choices. It has two terms: (n-1)^n from A065440(n), and (n-1)!^n from A091868(n-1). Such men's ranking tables having no two men with the same first choice arise in A343694, A343475, and A344663.

a(n)*A123234 is a useful alternative to A343696 which combines a Latin men's ranking table with an arbitrary women's table, since it gives fewer instances to consider.

a(n) is the number of reduced men's ranking tables in the stable marriage problem of order n. In the SMP (as noted in A351409), relabeling men or women has no effect on the number of stable matchings. So the women can be relabeled to normalize the order of man #1's rankings (with woman #1 as his first choice and woman n as his last choice), and then the men except man #1 can be relabeled to normalize the lexicographic order of those men's rankings. Since man #1's rankings end up fixed in natural order, they do not contribute to the number of possibilities, leaving n! multichoose (n-1) ways to arrange the rankings of the other n-1 men.

The number of unreduced men's ranking tables is given by A036740. Relabeling just the women reduces this to A134366. Alternately, relabeling just the men reduces A036740 to A344690. Relabeling both men and women reduces the men's relabeling reduction, A344690, by a factor of (n!+n-1)/n to a(n).

It might be tempting to try to reduce A344690 by a factor of n!, but that doesn't work because not all of man #1's rankings are equally likely after relabeling all the men to give man #1 the lexicographically least rankings.

There is room for further relabeling reduction from a(n), given by A263921. The reduction from a(n) to A263921 is analogous to the reduction from reduced latin squares, A000315, to A123234.

Each of the a(n) reduced men's ranking tables can be combined with the A036740 possible unreduced women's ranking tables to form complete instances, but these instances have more possibilities than A351409. For example, a(3)*A036740(3)=21*216=4536 > A351409(3)=3888. However, fewer possibilities result from using A263921 in place of a(n), although the men's ranking tables of A263921 may not be as straightforward to generate. With A263921(3)=10, 10*216=2160 < 3888.

In the Latin Stable Marriage Problem of order n, the sum of a man and woman's rankings of each other is n+1. This implies that the men's and women's ranking tables are Latin squares. As a subproblem of the Stable Marriage Problem, Latin instances provide lower bounds for the maximum number of stable matchings in the general problem, such as A005154 and A065982. For sizes 1 to 4, Latin instances provide exact bounds; they are conjectured to provide exact bounds for sizes a power of 2; they provide the best lower bounds known for sizes 6, 10, 12, and 24, of 48, 1000, 6472, and 126112960, respectively.

The next term, a(8), is conjectured to be 268, consistent with A005154. The minimum number of stable matchings for Latin instances of order n is n, and is realized for the cyclic group of order n. The average number of stable matchings is 7 for n=4 (cf. A351430 showing an average of about 1.5 for the general problem), and benefits from avoidance of mutual first choices and more generally the lack of overlap between the men's and women's preferred matchings. The Latin squares of A005154 and A065982 can be interpreted as multiplication tables of groups, n-th powers of the cyclic group C2 and n-th dihedral groups, respectively.

The sequence decreases from a(4)=10 to a(5)=9, in contrast to the corresponding sequence for the general problem, which Thurber showed to be strictly increasing. This has motivated the study of less restrictive subproblems, such as pseudo-Latin squares (A069124, A069156), Latin x Latin instances (A344664, A344665, A343697), instances where participants have different first choices (A343475, A343694, A343695), or instances with unspecified/tied/template rankings (A284458 with only first choices specified).

The sequence is empirically derived, originally based on reduced Latin squares (A000315). There are fewer instances to try using RC-equivalent Latin squares (A123234) instead of reduced Latin squares.