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A215297 T(n,k) = number of permutations of 0..floor((n*k-2)/2) on odd squares of an n X k array such that each row and column of odd squares is increasing.

%I A215297 #21 Mar 28 2019 10:45:50
%S A215297 1,1,1,1,2,1,1,3,3,1,1,6,6,6,1,1,10,30,30,10,1,1,20,70,280,70,20,1,1,
%T A215297 35,420,2100,2100,420,35,1,1,70,1050,23100,23100,23100,1050,70,1,1,
%U A215297 126,6930,210210,1051050,1051050,210210,6930,126,1,1,252,18018,2522520,14294280
%N A215297 T(n,k) = number of permutations of 0..floor((n*k-2)/2) on odd squares of an n X k array such that each row and column of odd squares is increasing.
%C A215297 Table starts
%C A215297 .1...1.....1........1...........1..............1.................1
%C A215297 .1...2.....3........6..........10.............20................35
%C A215297 .1...3.....6.......30..........70............420..............1050
%C A215297 .1...6....30......280........2100..........23100............210210
%C A215297 .1..10....70.....2100.......23100........1051050..........14294280
%C A215297 .1..20...420....23100.....1051050.......85765680........5703417720
%C A215297 .1..35..1050...210210....14294280.....5703417720......577185873264
%C A215297 .1..70..6930..2522520...814773960...577185873264...337653735859440
%C A215297 .1.126.18018.25729704.12547518984.48236247979920.43364386933948080
%C A215297 Even columns match A215292.
%C A215297 The first column is number of symmetric standard Young tableaux of shape (n), the second column is number of symmetric standard Young tableaux of shape (n,n) and the third column is number of symmetric standard Young tableaux of shape (n,n,n). - _Ran Pan_, May 21 2015
%H A215297 R. H. Hardin, <a href="/A215297/b215297.txt">Table of n, a(n) for n = 1..1000</a>
%H A215297 Hodge, Jonathan K.; Krines, Mark; Lahr, Jennifer, <a href="https://doi.org/10.1007/s11083-009-9112-1">Preseparable extensions of multidimensional preferences</a>, Order 26, No. 2, 125-147 (2009), Table 1.
%H A215297 Ran Pan, <a href="http://www.math.ucsd.edu/~projectp/warmups/eP.html">Exercise P</a>, <a href="http://www.math.ucsd.edu/~projectp/problems/p4.html">Problem 4</a>, Project P.
%F A215297 f1=floor(k/2), f2=floor((k+1)/2), f3=floor((n+1)/2), f4=floor(n/2);
%F A215297 T(n,k) = A060854(f1,f3)*A060854(f2,f4)*binomial(f1*f3+f2*f4,f1*f3).
%e A215297 Some solutions for n=5, k=4:
%e A215297 ..x..0..x..4....x..0..x..1....x..1..x..3....x..0..x..6....x..0..x..1
%e A215297 ..1..x..2..x....4..x..7..x....0..x..8..x....3..x..5..x....3..x..7..x
%e A215297 ..x..3..x..8....x..2..x..3....x..2..x..5....x..1..x..7....x..2..x..5
%e A215297 ..6..x..7..x....5..x..9..x....4..x..9..x....4..x..9..x....6..x..8..x
%e A215297 ..x..5..x..9....x..6..x..8....x..6..x..7....x..2..x..8....x..4..x..9
%Y A215297 Column 2 is A001405. Column 4 is A215288. Column 6 is A215290.
%K A215297 nonn,tabl
%O A215297 1,5
%A A215297 _R. H. Hardin_, Aug 07 2012