A214020 Number A(n,k) of n X k chess tableaux; square array A(n,k), n>=0, k>=0, read by antidiagonals.
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 2, 6, 6, 2, 1, 1, 1, 1, 0, 22, 0, 22, 0, 1, 1, 1, 1, 5, 92, 324, 324, 92, 5, 1, 1, 1, 1, 0, 422, 0, 8716, 0, 422, 0, 1, 1, 1, 1, 14, 2074, 47570, 343234, 343234, 47570, 2074, 14, 1, 1
Offset: 0
Examples
A(4,3) = A(3,4) = 6: [1 4 7] [1 4 5] [1 2 3] [1 4 7] [ 1 4 7] [ 1 2 3] [2 5 10] [2 7 10] [4 7 10] [2 5 10] [ 2 5 8] [ 4 5 6] [3 8 11] [3 8 11] [5 8 11] [3 6 11] [ 3 6 9] [ 7 8 9] [6 9 12] [6 9 12] [6 9 12] [8 9 12] [10 11 12] [10 11 12]. Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, 1, 1, ... 1, 1, 1, 1, 1, 1, 1, 1, ... 1, 1, 0, 1, 0, 2, 0, 5, ... 1, 1, 1, 2, 6, 22, 92, 422, ... 1, 1, 0, 6, 0, 324, 0, 47570, ... 1, 1, 2, 22, 324, 8716, 343234, 17423496, ... 1, 1, 0, 92, 0, 343234, 0, 8364334408, ... 1, 1, 5, 422, 47570, 17423496, 8364334408, 6873642982160, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..24, flattened
- T. Y. Chow, H. Eriksson and C. K. Fan, Chess tableaux, Elect. J. Combin., 11 (2) (2005), #A3.
- Jonas Sjöstrand, On the sign-imbalance of partition shapes, arXiv:math/0309231v3 [math.CO], 2005.
- Wikipedia, Young tableau
Programs
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Maple
b:= proc() option remember; local s; s:= add(i, i=args); `if`(s=0, 1, add(`if`(irem(s+i-args[i], 2)=1 and args[i]>`if`(i=nargs, 0, args[i+1]), b(subsop(i=args[i]-1, [args])[]), 0), i=1..nargs)) end: A:= (n, k)-> `if`(n -
Mathematica
b[args_List] := b[args] = Module[{s = Total[args], nargs = Length[args]}, If[s == 0, 1, Sum[If[Mod[s + i - args[[i]], 2] == 1 && args[[i]] > If[i == nargs, 0, args[[i + 1]]], b[ReplacePart[args, i -> args[[i]] - 1]], 0], {i, 1, nargs}]]]; A[n_, k_] := If[n < k, A[k, n], If[k < 2, 1, b[Array[n &, k]]]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jan 21 2015, after Alois P. Heinz *)
Comments