A341761 Triangle read by rows in which row n is the coefficients of the subword complexity polynomial S(n,x).
0, 0, 1, 0, -1, 3, 0, 0, -3, 6, 0, -1, 1, -6, 10, 0, 2, -6, 4, -10, 15, 0, -2, 10, -18, 10, -15, 21, 0, 2, -12, 31, -41, 20, -21, 28, 0, -1, 11, -41, 76, -80, 35, -28, 36, 0, 2, -6, 37, -109, 161, -141, 56, -36, 45, 0, 0, 9, -29, 110, -251, 308, -231, 84, -45, 55
Offset: 0
Examples
The triangle begins as 0; 0, 1; 0, -1, 3; 0, 0, -3, 6; 0, -1, 1, -6, 10; 0, 2, -6, 4, -10, 15; 0, -2, 10, -18, 10, -15, 21; 0, 2, -12, 31, -41, 20, -21, 28; ... Below lists some subword complexity polynomials: S(0,x) = 0 S(1,x) = 1*x S(2,x) = -1*x + 3*x^2 S(3,x) = -3*x^2 + 6*x^3 S(4,x) = -1*x + x^2 - 6*x^3 + 10*x^4 ... For n = 3 and x = 2 there are eight possible words: "aaa", "aab", "aba", "abb", "baa", "bab", "bba" and "bbb", and their subword complexities are 3, 5, 5, 5, 5, 5, 5 and 3 respectively, and their sum = S(3,2) = -3*(2^2)+6*(2^3) = 36.
Links
- Shiyao Guo, Table of n, a(n) for n = 0..1890
- Shiyao Guo, On the Expected Subword Complexity of Random Words.
- Shiyao Guo, C++ program that computes subword complexity polynomial for n up to 60.
Crossrefs
Cf. A340885 (values of S(n,2)).
Programs
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Mathematica
S[n_, x_] := Total[Length /@ DeleteDuplicates /@ Subsequences /@ Tuples[Table[i, {i, 0, x}], n] - 1]; A341761[n_] := CoefficientList[FindSequenceFunction[ParallelTable[S[n, i], {i, 0, n + 1}], x], {x}]; Join[{0, 0, 1}, Table[A341761[n], {n, 2, 7}] // Flatten] (* Robert P. P. McKone, Feb 20 2021 *)
Comments