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

Author

Alois P. Heinz, Feb 29 2012

Keywords

• nonn
• tabl
• walk

Comments

A(n,k) is also the number of (n*k-1)-step walks on k-dimensional cubic lattice from (1,0,...,0) to (n,n,...,n) with positive unit steps in all dimensions such that for each point (p_1,p_2,...,p_k) we have p_1<=p_2<=...<=p_k or p_1>=p_2>=...>=p_k.

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

Rows 0+1, 2-10 give: A000012, A141351 (for n>1), A208616, A208617, A208618, A208619, A208620, A208621, A208622, A208623.

Columns 0+1, 2-10 give: A000012, A088218, A185148, A208624, A208625, A208626, A208627, A208628, A208629, A208630.

Main diagonal gives: A208631.

Antidiagonal sums give: A208729.

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 *)