OFFSET
1,2
REFERENCES
Stanley, R., Enumerative Combinatorics, Vol. 2, Proposition 7.10.3 and Vol. 1, Sec 3.5 Chains in Distributive Lattices.
LINKS
Alois P. Heinz, Antidiagonals n = 1..12, flattened
EXAMPLE
Square array A(n,m) begins:
1, 2, 5, 14, ...
2, 48, 2452, 183958, ...
5, 2452, 4877756, 20071150430, ...
14, 183958, 20071150430, 6708527580006468, ...
MAPLE
b := proc(l) option remember; local n; n:= nops(l);
`if`({seq(l[i][], i=1..n)}={0}, 1, add(`if`(l[i][1]>l[i][2] and
l[i][1]>l[i+1][1], b(subsop(i=[l[i][1]-1, l[i][2]], l)), 0),
i=1..n-1)+ add(`if`(l[i][2]>l[i+1][2], b(subsop(i=[l[i][1],
l[i][2]-1], l)), 0), i=1..n-1)+ `if`(l[n][1]>l[n][2],
b(subsop(n=[l[n][1]-1, l[n][2]], l)), 0)+ `if`(l[n][2]>0,
b(subsop(n=[l[n][1], l[n][2]-1], l)), 0))
end:
A:= (n, m)-> `if`(m>=n, b([[m$2]$n]), b([[n$2]$m])):
seq(seq(A(n, d+1-n), n=1..d), d=1..8); # Alois P. Heinz, Jun 29 2012
MATHEMATICA
b[l_List] := b[l] = With[{n = Length[l]}, If[Union[Table[l[[i]], {i, 1, n}] // Flatten] == {0}, 1, Sum[If[l[[i, 1]] > l[[i, 2]] && l[[i, 1]] > l[[i+1, 1]], b[ReplacePart[l, i -> {l[[i, 1]]-1, l[[i, 2]]}]], 0], {i, 1, n-1}] + Sum[If[l[[i, 2]] > l[[i+1, 2]], b[ReplacePart[l, i -> {l[[i, 1]], l[[i, 2]]-1}]], 0], {i, 1, n-1}] + If[l[[n, 1]] > l[[n, 2]], b[ReplacePart[l, n -> {l[[n, 1]]-1, l[[n, 2]]} ]], 0] + If[l[[n, 2]] > 0, b[ReplacePart[l, n -> {l[[n, 1]], l[[n, 2]]-1}]], 0]]] ; A[n_, m_] := If[m >= n, b[Array[{m, m}&, n]], b[Array[{n, n}&, m]]]; Table[ Table[A[n, d+1-n], {n, 1, d}], {d, 1, 8}] // Flatten (* Jean-François Alcover, Mar 11 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
AUTHOR
Mitch Harris, Dec 27 2005
EXTENSIONS
Edited by Alois P. Heinz, Jun 29 2012
STATUS
approved