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

1, 1, 4, 29, 290, 3532, 49100, 750325, 12310294, 213446666, 3868253164, 72686739116, 1407643591804, 27964937748724, 567853691242796, 11751537336221989, 247263499985110046, 5279409371079693454, 114199628255736623996, 2499214354674134770354

Offset

1,3

Comments

Row 3 of A181196.

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.

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.)

a(n) is also the number of maximum packings of pattern

[5 6]

[3 4]

[1 2] in column-strict arrays of size 3 X n+1. - Ran Pan, Apr 13 2015

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

Links

Alois P. Heinz, Table of n, a(n) for n = 1..220

Joerg Arndt, The a(4)=29 truncated shifted standard Young tableaux of shape [4,4,4].

J. B. Lewis, Pattern Avoidance for Alternating Permutations and Reading Words of Tableaux, Ph. D. Dissertation, Department of Mathematics, MIT, 2012. - From N. J. A. Sloane, Oct 12 2012

Ran Pan, Problem 0, Project P.

Ping Sun, Enumeration of standard Young tableaux of shifted strips with constant width, arXiv:1506.07256 [math.CO], 24 Jun 2015.

Formula

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.

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

a(n) ~ 3^(3*n+1/2)/(64*Pi*n^4). - Vaclav Kotesovec, Jul 21 2013

Example

All four 3 X 3 examples:
1..2..3....1..2..3....1..2..4....1..2..4
4..5..6....4..5..7....3..5..6....3..5..7
7..8..9....6..8..9....7..8..9....6..8..9

Maple

a:= n-> `if`(n<2, 1, add(((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)):
seq(a(n), n=1..30);  # Alois P. Heinz, Jul 01 2012

Mathematica

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 *)

Crossrefs

Row n=3 of A227578. - Alois P. Heinz, Jul 17 2013

Sequence in context: A083072 A302586 A302609 * A217807 A348651 A360584

Adjacent sequences: A181194 A181195 A181196 * A181198 A181199 A181200

Keyword

nonn

Author

R. H. Hardin, Oct 10 2010

Extensions

Formula and comments from Joel B. Lewis, Jul 25 2011

Status

approved