login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

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.
10
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
KEYWORD
nonn,tabl
AUTHOR
R. H. Hardin, Oct 10 2010
STATUS
approved