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
A227578 is a similar but different array.
KEYWORD
nonn,tabl
AUTHOR
R. H. Hardin, Oct 10 2010
STATUS
approved