cp's OEIS Frontend

A181197 Number of 3 X n matrices containing a permutation of 1..3*n in increasing order rowwise, columnwise and (downwards) antidiagonally.

%I A181197 #47 Dec 31 2023 10:58:15
%S A181197 1,1,4,29,290,3532,49100,750325,12310294,213446666,3868253164,
%T A181197 72686739116,1407643591804,27964937748724,567853691242796,
%U A181197 11751537336221989,247263499985110046,5279409371079693454,114199628255736623996,2499214354674134770354
%N A181197 Number of 3 X n matrices containing a permutation of 1..3*n in increasing order rowwise, columnwise and (downwards) antidiagonally.
%C A181197 Row 3 of A181196.
%C A181197 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 A181197 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 A181197 a(n) is also the number of maximum packings of pattern
%C A181197 [5 6]
%C A181197 [3 4]
%C A181197 [1 2] in column-strict arrays of size 3 X n+1. - _Ran Pan_, Apr 13 2015
%C A181197 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 A181197 Alois P. Heinz, <a href="/A181197/b181197.txt">Table of n, a(n) for n = 1..220</a>
%H A181197 Joerg Arndt, <a href="/A181197/a181197.txt">The a(4)=29 truncated shifted standard Young tableaux of shape [4,4,4]</a>.
%H A181197 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 A181197 Ran Pan, <a href="http://www.math.ucsd.edu/~projectp/problems/p0.html">Problem 0</a>, Project P.
%H A181197 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 A181197 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 A181197 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 A181197 a(n) ~ 3^(3*n+1/2)/(64*Pi*n^4). - _Vaclav Kotesovec_, Jul 21 2013
%e A181197 All four 3 X 3 examples:
%e A181197 1..2..3....1..2..3....1..2..4....1..2..4
%e A181197 4..5..6....4..5..7....3..5..6....3..5..7
%e A181197 7..8..9....6..8..9....7..8..9....6..8..9
%p A181197 a:= n-> `if`(n<2, 1, add(((2*n+k-1)!*(n-k)*(n-k-1)) /
%p A181197          (n!*(n-1)!*k!*(2*n-1)*(n+k)*(n+k-1)), k=0..n-2)):
%p A181197 seq(a(n), n=1..30);  # _Alois P. Heinz_, Jul 01 2012
%t A181197 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 A181197 Row n=3 of A227578. - _Alois P. Heinz_, Jul 17 2013
%K A181197 nonn
%O A181197 1,3
%A A181197 _R. H. Hardin_, Oct 10 2010
%E A181197 Formula and comments from _Joel B. Lewis_, Jul 25 2011