A369326 Array read by ascending antidiagonals: A(n,k) is the number of words of length n over the alphabet [k] and sortable by a (2,1)-pop stack of depth 2.
1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 8, 9, 4, 1, 0, 1, 16, 24, 16, 5, 1, 0, 1, 32, 59, 52, 25, 6, 1, 0, 1, 64, 138, 149, 95, 36, 7, 1, 0, 1, 128, 313, 396, 310, 156, 49, 8, 1, 0, 1, 256, 696, 1003, 923, 571, 238, 64, 9, 1, 0, 1, 512, 1527, 2458, 2585, 1884, 966, 344, 81, 10, 1
Offset: 0
Examples
The array begins: 1, 1, 1, 1, 1, 1, ... 0, 1, 2, 3, 4, 5, ... 0, 1, 4, 9, 16, 25, ... 0, 1, 8, 24, 52, 95, ... 0, 1, 16, 59, 149, 310, ... 0, 1, 32, 138, 396, 923, ... ...
Links
- Toufik Mansour, Howard Skogman, and Rebecca Smith, Sorting inversion sequences, arXiv:2401.06662 [math.CO], 2024. See Theorem 3.25 at page 13.
Crossrefs
Programs
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Mathematica
A[n_,k_]:=SeriesCoefficient[((1-x)(1-2x)-((1-x)(1-2x)+x^2)y)/((1-x)(1-2x)-(1-x)(2-3x)y+(1-2x)y^2),{x,0,n},{y,0,k}]; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten
Formula
G.f.: ((1 - x)(1 - 2*x) - ((1 - x)*(1 - 2*x) + x^2)*y)/((1 - x)*(1 - 2*x) - (1 - x)*(2 - 3*x)*y + (1 - 2*x)*y^2).