A372873 Triangle read by rows: T(n,k) is the number of flattened Catalan words of length n with exactly k runs of descents.
1, 0, 2, 0, 1, 4, 0, 0, 6, 8, 0, 0, 1, 24, 16, 0, 0, 0, 10, 80, 32, 0, 0, 0, 1, 60, 240, 64, 0, 0, 0, 0, 14, 280, 672, 128, 0, 0, 0, 0, 1, 112, 1120, 1792, 256, 0, 0, 0, 0, 0, 18, 672, 4032, 4608, 512, 0, 0, 0, 0, 0, 1, 180, 3360, 13440, 11520, 1024
Offset: 1
Examples
The triangle begins: 1; 0, 2; 0, 1, 4; 0, 0, 6, 8; 0, 0, 1, 24, 16; 0, 0, 0, 10, 80, 32; 0, 0, 0, 1, 60, 240, 64; 0, 0, 0, 0, 14, 280, 672, 128; 0, 0, 0, 0, 1, 112, 1120, 1792, 256; ... T(4,3) = 6 since there 6 flattened Catalan words of length 4 with 3 runs of descents: 0010, 0100, 0101, 0110, 0120, and 0121.
Links
- Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See p. 11.
Programs
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Mathematica
T[n_,k_]:=SeriesCoefficient[x*y*(1-2*x*y)/(1-4*x*y-x^2*y+4x^2*y^2),{x,0,n},{y,0,k}]; Table[T[n,k],{n,11},{k,n}]//Flatten (* or *) T[n_,k_]:=2^(2*k-n-1)*Binomial[n-1, 2*(n-k)]; Table[T[n,k],{n,11},{k,n}]//Flatten