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 A344692 Irregular triangle read by rows in which T(n,k) is the number of stable matchings in the stable marriage problem with n men and n women such that there exists a stable matching with an egalitarian cost of k. 1

%I #47 Jan 27 2022 20:40:35

%S 0,1,0,0,0,2,8,8,0,0,0,0,0,384,2304,7488,14592,18072,13104,4380,0,0,0,

%T 0,0,0,0,40310784,322486272,1397440512,4299816960,10080681984,

%U 18632540160,27586068480,32664453120,30544625664,21941452800,11480334336,3963617280,707788800,0

%N Irregular triangle read by rows in which T(n,k) is the number of stable matchings in the stable marriage problem with n men and n women such that there exists a stable matching with an egalitarian cost of k.

%C The egalitarian cost of a stable matching is the sum of the mutual rankings of the people in couples.

%C Each preference profile with n men and women that has m different stable matchings with an egalitarian cost of k contributes m to T(n,k). The sequence that counts these m matchings as one is A344691.

%C The lowest and, therefore, optimal mutual ranking of two people is 2, which occurs when they rank each other first. Thus, the smallest possible egalitarian cost of a stable matching with n men and n women is 2*n. So, for k < 2*n, T(n,k) = 0.

%e Triangle begins:

%e 0, 1;

%e 0, 0, 0, 2, 8, 8;

%e 0, 0, 0, 0, 0, 384, 2304, 7488, 14592, 18072, 13104, 4380;

%e 0, 0, 0, 0, 0, 0, 0, 40310784, 322486272, 1397440512, ... ;

%e ...

%e The n-th row starts with 2*n-1 zeros.

%e The total number of terms in row 3 and 4 is 12 and 20 respectively.

%e If two people rank each other first, they are called soulmates. Therefore, if the egalitarian cost is 2*n then there are n pairs of soulmates.

%e A343698(n) counts preference profiles with n men and n women that have n pairs of soulmates. Moreover, if we have n pairs of soulmates in a profile, there's only one stable matching with egalitarian cost 2n. Thus, we have T(n,2n) = A343698(n).

%e If n = 2 and k = 4, we have two pairs of soulmates. There are two preference profiles like this. In the first profile, the first man and the first woman are soulmates as well as the second man and the second woman. In the second profile, the first man and the second woman as well as the second man and the first woman are soulmates. Thus T(2,4) = 2.

%Y Cf. A185141, A343698, A344691.

%K nonn,tabf

%O 1,6

%A _Tanya Khovanova_ and MIT PRIMES STEP Senior group, Jun 22 2021

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