cp's OEIS Frontend

A383370 Number of partial orders on {1,2,...,n} that are contained in the usual linear order, whose dual is given by the relabelling k -> n+1-k.

%I A383370 #14 May 02 2025 12:50:40
%S A383370 1,1,2,3,12,25,172,482,5318,19675,333768,1609846,40832554,254370640,
%T A383370 9459449890,75546875426,4061670272088
%N A383370 Number of partial orders on {1,2,...,n} that are contained in the usual linear order, whose dual is given by the relabelling k -> n+1-k.
%C A383370 a(n) is the number of n X n upper triangular Boolean matrices B with all diagonal entries 1 such that B = B^2, which are symmetric about the antidiagonal. These matrices can be seen as closed sets of inversions (pairs (i,j) with 1 <= i < j <= n). A set of inversions E is closed if for all i < j < k, if E contains (i,j) and (j,k) then it contains (i,k).
%e A383370 The Boolean matrices corresponding to a(4) = 12:
%e A383370   1 0 0 0    1 0 0 1    1 0 0 0    1 0 0 1
%e A383370   0 1 0 0    0 1 0 0    0 1 1 0    0 1 1 0
%e A383370   0 0 1 0    0 0 1 0    0 0 1 0    0 0 1 0
%e A383370   0 0 0 1    0 0 0 1    0 0 0 1    0 0 0 1
%e A383370 .
%e A383370   1 0 1 0    1 0 1 1    1 0 1 0    1 0 1 1
%e A383370   0 1 0 1    0 1 0 1    0 1 1 1    0 1 1 1
%e A383370   0 0 1 0    0 0 1 0    0 0 1 0    0 0 1 0
%e A383370   0 0 0 1    0 0 0 1    0 0 0 1    0 0 0 1
%e A383370 .
%e A383370   1 1 0 0    1 1 0 1    1 1 1 1    1 1 1 1
%e A383370   0 1 0 0    0 1 0 0    0 1 0 1    0 1 1 1
%e A383370   0 0 1 1    0 0 1 1    0 0 1 1    0 0 1 1
%e A383370   0 0 0 1    0 0 0 1    0 0 0 1    0 0 0 1
%o A383370 (SageMath)
%o A383370 def a(n):
%o A383370     S = set()
%o A383370     for P in Posets(n):
%o A383370         if P.is_isomorphic(P.dual()):
%o A383370             for l in P.linear_extensions():
%o A383370                 t = tuple(tuple(int(P.is_lequal(l[j],l[i])) for j in range(i)) for i in range(1,len(l)))
%o A383370                 if all(t[j][i]==t[n-i-2][n-j-2] for i in range((n-1)//2) for j in range(i,n-i-2)):
%o A383370                     S.add(t)
%o A383370     return len(S)
%Y A383370 Cf. A006455, A037223.
%K A383370 nonn,hard,more
%O A383370 0,3
%A A383370 _Ludovic Schwob_, Apr 24 2025
%E A383370 a(10)-a(16) from _Christian Sievers_, May 02 2025