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%I #6 May 20 2024 10:30:15
%S 1,4,32,400,6660,138576,3460464,100821904,3357239440
%N Number of (0,1)-matrices with sum of entries equal to 2*n or 2*n+1, no zero rows or columns, which are centrally symmetric.
%C a(n) is the number ordered pairs of semistandard Young tableaux of size 2*n or 2*n+1 with consecutive entries (i.e., if i is in T, and 1<=j<=i, then j is in T) and with conjugate shapes, which are both invariant under Schützenberger involution.
%e The a(2) = 32 matrices with sum of entries equal to 4:
%e [1 0 0 0] [0 0 0 1] [1 0 0 0] [0 0 0 1] [0 0 1 0] [0 1 0 0]
%e [0 1 0 0] [0 1 0 0] [0 0 1 0] [0 0 1 0] [0 0 0 1] [1 0 0 0]
%e [0 0 1 0] [0 0 1 0] [0 1 0 0] [0 1 0 0] [1 0 0 0] [0 0 0 1]
%e [0 0 0 1] [1 0 0 0] [0 0 0 1] [1 0 0 0] [0 1 0 0] [0 0 1 0]
%e .
%e [0 1 0 0] [0 0 1 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1]
%e [0 0 0 1] [1 0 0 0] [0 1 0] [1 0 0] [0 0 1] [0 1 0]
%e [1 0 0 0] [0 0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0]
%e [0 0 1 0] [0 1 0 0] [0 0 1] [0 1 0] [0 1 0] [1 0 0]
%e .
%e [1 0] [0 1] [1 0] [0 1] [1]
%e [1 0] [0 1] [0 1] [1 0] [1]
%e [0 1] [1 0] [1 0] [0 1] [1]
%e [0 1] [1 0] [0 1] [1 0] [1]
%e .
%e [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] [0 1 0] [1 0] [0 1]
%e [0 1 1 0] [1 0 0 1] [1 0 0 1] [0 1 1 0] [1 0 1] [1 1] [1 1]
%e [0 0 0 1] [0 0 1 0] [0 1 0 0] [1 0 0 0] [0 1 0] [0 1] [1 0]
%e .
%e [1 1 0 0] [0 0 1 1] [1 0 1 0] [0 1 0 1] [1 1 0] [0 1 1] [1 1]
%e [0 0 1 1] [1 1 0 0] [0 1 0 1] [1 0 1 0] [0 1 1] [1 1 0] [1 1]
%e .
%e [1 1 1 1]
%Y Cf. A275787, A370396.
%K nonn,more
%O 0,2
%A _Ludovic Schwob_, May 20 2024
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