A208615 Number of Young tableaux A(n,k) with n k-length rows, increasing entries down the columns and monotonic entries along the rows (first row increasing); square array A(n,k), n>=0, k>=0, read by antidiagonals.
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 6, 10, 1, 1, 1, 1, 15, 53, 35, 1, 1, 1, 1, 43, 491, 587, 126, 1, 1, 1, 1, 133, 6091, 25187, 7572, 462, 1, 1, 1, 1, 430, 87781, 1676707, 1725819, 109027, 1716, 1, 1, 1, 1, 1431, 1386529, 140422657, 705002611, 144558247, 1705249, 6435, 1, 1
Offset: 0
Examples
A(2,3) = 6: +---+ +---+ +---+ +---+ +---+ +---+ |123| |123| |124| |125| |134| |135| |456| |654| |356| |346| |256| |246| +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ |x |100| |x |100| |x |100| |x |100| |x |100| |x |100| | x |110| | x |110| | x |110| | x |110| |x |200| |x |200| | x|111| | x|111| |x |210| |x |210| | x |210| | x |210| |x |211| | x|112| | x|211| | x |220| | x|211| | x |220| | x |221| | x |122| | x |221| | x|221| | x |221| | x|221| | x|222| |x |222| | x|222| | x|222| | x|222| | x|222| +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, 1, ... 1, 1, 1, 1, 1, 1, 1, ... 1, 1, 3, 6, 15, 43, 133, ... 1, 1, 10, 53, 491, 6091, 87781, ... 1, 1, 35, 587, 25187, 1676707, 140422657, ... 1, 1, 126, 7572, 1725819, 705002611, 396803649991, ... 1, 1, 462, 109027, 144558247, 398084427253, 1672481205752413, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..25, flattened
Crossrefs
Programs
-
Maple
b:= proc() option remember; `if`(nargs<2, 1, `if`(args[1]=args[nargs], `if`(args[1]=0, 1, 2* b(args[1]-1, seq(args[i], i=2..nargs))), `if`(args[1]>0, b(args[1]-1, seq(args[i], i=2..nargs)), 0) +add(`if`(args[j]>args[j-1], b(seq(args[i] -`if`(i=j, 1, 0) , i=1..nargs)), 0), j=2..nargs) )) end: A:= (n, k)-> `if`(n=0 or k=0, 1, b(n-1, n$(k-1))): seq(seq(A(n, d-n), n=0..d), d=0..12); -
Mathematica
b[args__] := b[args] = If[(nargs = Length[{args}]) < 2, 1, If[First[{args}] == Last[{args}], If[First[{args}] == 0, 1, 2*b[First[{args}]-1, Sequence @@ Rest[{args}]]], If[First[{args}] > 0, b[First[{args}]-1, Sequence @@ Rest[{args}]], 0] + Sum [If[{args}[[j]] > {args}[[j-1]], b[Sequence @@ Table[{args}[[i]] - If[i == j, 1, 0], {i, 1, nargs}]], 0], {j, 2, nargs}] ] ]; a[n_, k_] := If[n == 0 || k == 0, 1, b[n-1, Sequence @@ Array[n&, k-1]]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)
Comments