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%I #47 Dec 31 2023 10:58:15
%S 1,1,4,29,290,3532,49100,750325,12310294,213446666,3868253164,
%T 72686739116,1407643591804,27964937748724,567853691242796,
%U 11751537336221989,247263499985110046,5279409371079693454,114199628255736623996,2499214354674134770354
%N Number of 3 X n matrices containing a permutation of 1..3*n in increasing order rowwise, columnwise and (downwards) antidiagonally.
%C Row 3 of A181196.
%C Equivalently, the number of "truncated shifted standard Young tableaux" of shape <n, n, n>; in other words, if we shift the middle row to the right by one unit and the bottom row to the right by two units, we require that the resulting diagram be increasing as we read down or to the right.
%C To count these tableaux, observe that if we put the entry 2n + 2 + k in the last position of the second row, the bottom row must end with the entries 2n + 3 + k, ..., 3n. The remaining figure can be filled in arbitrarily; it is a shifted Young diagram of shape <n, n - 1, k>. Now apply the hook-length formula for shifted Young tableaux. (This argument is due to Greta Panova.)
%C a(n) is also the number of maximum packings of pattern
%C [5 6]
%C [3 4]
%C [1 2] in column-strict arrays of size 3 X n+1. - _Ran Pan_, Apr 13 2015
%C a(n) is also the number of standard Young tableaux of shape (n,n,n) (French notation) such that for any element T(i,j) in the tableau T, its upper element T(i+1,j) is larger than its right element T(i,j+1). - _Ran Pan_, Apr 13 2015
%H Alois P. Heinz, <a href="/A181197/b181197.txt">Table of n, a(n) for n = 1..220</a>
%H Joerg Arndt, <a href="/A181197/a181197.txt">The a(4)=29 truncated shifted standard Young tableaux of shape [4,4,4]</a>.
%H J. B. Lewis, <a href="http://hdl.handle.net/1721.1/73444">Pattern Avoidance for Alternating Permutations and Reading Words of Tableaux</a>, Ph. D. Dissertation, Department of Mathematics, MIT, 2012. - From _N. J. A. Sloane_, Oct 12 2012
%H Ran Pan, <a href="http://www.math.ucsd.edu/~projectp/problems/p0.html">Problem 0</a>, Project P.
%H Ping Sun, <a href="http://arxiv.org/abs/1506.07256">Enumeration of standard Young tableaux of shifted strips with constant width</a>, arXiv:1506.07256 [math.CO], 24 Jun 2015.
%F a(n) = Sum_{k=0..n-2} ((2n+k-1)!*(n-k)*(n-k-1)) / (n!*(n-1)!*k!*(2n-1) * (n+k)*(n+k-1)) for n>=2, a(1) = 1.
%F Recurrence: (2*n-1)*(7*n-13)*n^2*a(n) = 2*(182*n^4 - 1185*n^3 + 2722*n^2 - 2625*n + 900)*a(n-1) + 3*(2*n-5)*(3*n-5)*(3*n-4)*(7*n-6)*a(n-2). - _Vaclav Kotesovec_, Jul 21 2013
%F a(n) ~ 3^(3*n+1/2)/(64*Pi*n^4). - _Vaclav Kotesovec_, Jul 21 2013
%e All four 3 X 3 examples:
%e 1..2..3....1..2..3....1..2..4....1..2..4
%e 4..5..6....4..5..7....3..5..6....3..5..7
%e 7..8..9....6..8..9....7..8..9....6..8..9
%p a:= n-> `if`(n<2, 1, add(((2*n+k-1)!*(n-k)*(n-k-1)) /
%p (n!*(n-1)!*k!*(2*n-1)*(n+k)*(n+k-1)), k=0..n-2)):
%p seq(a(n), n=1..30); # _Alois P. Heinz_, Jul 01 2012
%t Flatten[{1,Table[Sum[((2*n+k-1)!*(n-k)*(n-k-1))/(n!*(n-1)!*k!*(2*n-1)*(n+k)*(n+k-1)),{k,0,n-2}],{n,2,20}]}] (* _Vaclav Kotesovec_, Jul 21 2013 *)
%Y Row n=3 of A227578. - _Alois P. Heinz_, Jul 17 2013
%K nonn
%O 1,3
%A _R. H. Hardin_, Oct 10 2010
%E Formula and comments from _Joel B. Lewis_, Jul 25 2011
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