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A181196 T(n,k) = number of n X k matrices containing a permutation of 1..n*k in increasing order rowwise, columnwise, diagonally and (downwards) antidiagonally. 9
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 4, 1, 1, 1, 14, 29, 8, 1, 1, 1, 42, 290, 169, 16, 1, 1, 1, 132, 3532, 6392, 985, 32, 1, 1, 1, 429, 49100, 352184, 141696, 5741, 64, 1, 1, 1, 1430, 750325, 25097600, 36372976, 3142704, 33461, 128, 1, 1, 1, 4862, 12310294 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

Table starts:

.1.1...1......1..........1..............1...................1

.1.1...2......5.........14.............42.................132

.1.1...4.....29........290...........3532...............49100

.1.1...8....169.......6392.........352184............25097600

.1.1..16....985.....141696.......36372976.........14083834704

.1.1..32...5741....3142704.....3777546912.......8092149471168

.1.1..64..33461...69705920...392658046912....4673805856338368

.1.1.128.195025.1546100352.40820345224064.2702482348019033600

LINKS

R. H. Hardin and Alois P. Heinz, Antidiagonals n = 1..30, flattened

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

FORMULA

Empirical column 1: a(n) = a(n-1).

Empirical column 2: a(n) = a(n-1).

Empirical column 3: a(n) = 2*a(n-1).

Empirical column 4: a(n) = 6*a(n-1)-a(n-2).

Empirical column 5: a(n) = 24*a(n-1)-40*a(n-2)-8*a(n-3).

Empirical column 6: a(n) = 120*a(n-1)-1672*a(n-2)+544*a(n-3)-6672*a(n-4) +256*a(n-5).

Empirical column 7: a(n) = 720*a(n-1) -84448*a(n-2) +1503360*a(n-3) -17912224*a(n-4) -318223104*a(n-5) +564996096*a(n-6) +270471168*a(n-7) -11373824*a(n-8) +65536*a(n-9).

EXAMPLE

All solutions for 3 X 4:

..1..2..3..4....1..2..3..4....1..2..3..4....1..2..3..4....1..2..3..4

..5..6..7..8....5..6..7..9....5..6..7.10....5..6..8..9....5..6..8.10

..9.10.11.12....8.10.11.12....8..9.11.12....7.10.11.12....7..9.11.12

...

..1..2..3..6....1..2..3..6....1..2..3..6....1..2..3..6....1..2..3..6

..4..5..7..8....4..5..7..9....4..5..7.10....4..5..8..9....4..5..8.10

..9.10.11.12....8.10.11.12....8..9.11.12....7.10.11.12....7..9.11.12

...

..1..2..4..6....1..2..4..6....1..2..4..6....1..2..4..6....1..2..4..6

..3..5..7..8....3..5..7..9....3..5..7.10....3..5..8..9....3..5..8.10

..9.10.11.12....8.10.11.12....8..9.11.12....7.10.11.12....7..9.11.12

...

..1..2..3..5....1..2..3..5....1..2..3..5....1..2..3..5....1..2..3..5

..4..6..7..8....4..6..7..9....4..6..7.10....4..6..8..9....4..6..8.10

..9.10.11.12....8.10.11.12....8..9.11.12....7.10.11.12....7..9.11.12

...

..1..2..4..5....1..2..4..5....1..2..4..5....1..2..4..5....1..2..4..5

..3..6..7..8....3..6..7..9....3..6..7.10....3..6..8..9....3..6..8.10

..9.10.11.12....8.10.11.12....8..9.11.12....7.10.11.12....7..9.11.12

...

..1..2..3..7....1..2..3..7....1..2..4..7....1..2..4..7

..4..5..8..9....4..5..8.10....3..5..8..9....3..5..8.10

..6.10.11.12....6..9.11.12....6.10.11.12....6..9.11.12

MAPLE

b:= proc(l) option remember; local n; n:= nops(l);

      `if`({l[]}={0}, 1, add(`if`((i=1 or l[i-1]<=l[i]) and l[i]>

      `if`(i=n, 0, l[i+1]), b(subsop(i=l[i]-1, l)), 0), i=1..n))

    end:

T:= (n, k)-> b([n$k]):

seq(seq(T(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Jul 24 2012

MATHEMATICA

b[l_List] := b[l] = With[{n = Length[l]}, If[Union[l] == {0}, 1, Sum[If[(i == 1 || l[[i-1]] <= l[[i]]) && l[[i]] > If[i == n, 0, l[[i+1]]], b[ReplacePart[l, i -> l[[i]]-1]], 0], {i, 1, n}]]]; T[n_, k_] := b[Array[n&, k]]; Table[Table[T[n, 1+d-n], {n, 1, d}], {d, 1, 12}] // Flatten (* Jean-Fran├žois Alcover, Mar 06 2015, after Alois P. Heinz *)

CROSSREFS

Rows n=1-5 give: A000012, A000108, A181197, A181198, A181199.

Columns 1+2, 3-8 give: A000012, A011782, A001653, A181192, A181193, A181194, A181195.

Sequence in context: A121391 A241194 A008326 * A227578 A181783 A121395

Adjacent sequences:  A181193 A181194 A181195 * A181197 A181198 A181199

KEYWORD

nonn,tabl

AUTHOR

R. H. Hardin Oct 10 2010

STATUS

approved

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Last modified April 23 13:25 EDT 2021. Contains 343204 sequences. (Running on oeis4.)