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

Author

R. H. Hardin, Oct 10 2010

Keywords

• nonn
• tabl

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

Examples

			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
		

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.

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.

Programs

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 *)

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) [phone]*a(n-6) [phone]*a(n-7) -11373824*a(n-8) +65536*a(n-9).