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.

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

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

Brian T. Chan, Periodic P-Partitions, arXiv:1803.05594 [math.CO], 2018-2020.

Ping Sun, Enumeration of standard Young tableaux of shifted strips with constant width, El. J. Comb., 24 (2017), #P2.41; arXiv:1506.07256 [math.CO], 2015.

Antonio Vera López, Luis Martínez, Antonio Vera Pérez, Beatriz Vera Pérez and Olga Basova, Combinatorics related to Higman's conjecture I: Parallelogramic digraphs and dispositions, Linear Algebra and its Applications, Volume 530, 1 October 2017, p. 414-444. See Table 1.

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-François 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.

A227578 is a similar but different array.

Sequence in context: A241194 A352893 A008326 * A227578 A181783 A121395

Adjacent sequences: A181193 A181194 A181195 * A181197 A181198 A181199

Keyword

nonn,tabl

Author

R. H. Hardin, Oct 10 2010

Status

approved