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

0,13

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.

Links

Alois P. Heinz, Antidiagonals n = 0..25, flattened

Example

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, ...

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 *)

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.

Sequence in context: A208673 A010122 A220693 * A058663 A124371 A147989

Adjacent sequences: A208612 A208613 A208614 * A208616 A208617 A208618

Keyword

nonn,tabl,walk

Author

Alois P. Heinz, Feb 29 2012

Status

approved