A128249 T(n,k) is the number of unlabeled acyclic single-source automata with n transient states on a (k+1)-letter input alphabet.
1, 3, 1, 16, 7, 1, 127, 139, 15, 1, 1363, 5711, 1000, 31, 1, 18628, 408354, 189035, 6631, 63, 1, 311250, 45605881, 79278446, 5470431, 42196, 127, 1, 6173791, 7390305396, 63263422646, 12703473581, 147606627, 262459, 255, 1, 142190703, 1647470410551
Offset: 1
Links
- David Callan, A determinant of Stirling Cycle Numbers Count Unlabeled Acyclic Single-Source Automata math.CO/0704.0004.
- Manosij Ghosh Dastidar and Michael Wallner, Asymptotics of relaxed k-ary trees, arXiv:2404.08415 [math.CO], 2024. See p. 1.4.
Programs
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Maple
T := proc(n,k) local kn,A,i,j ; kn := k*n ; A := matrix(kn,kn) ; for i from 1 to kn do for j from 1 to kn do A[i,j] := abs(combinat[stirling1](floor((i-1)/k)+2,floor((i-1)/k)+1+i-j)) ; od ; od ; linalg[det](A) ; end: for d from 1 to 9 do for n from d to 1 by -1 do k := d+1-n ; printf("%d, ",T(n,k)) ; od ; od; -
Mathematica
t[n_, k_] := Module[{kn, a, i, j}, kn = k*n; For[i = 1, i <= kn, i++, For[j = 1, j <= kn, j++, a[i, j] = Abs[StirlingS1[Floor[(i-1)/k]+2, Max[0, Floor[(i-1)/k]+1+i-j]]]]]; Det[Array[a, {kn, kn}]]]; Table[t[n-k, k], {n, 1, 10}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Jan 10 2014, translated from Maple *)
Comments