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

0,6

Comments

S(n,x) is the sum of subword complexities (number of nonempty distinct subwords) of all words of length n and an alphabet with size x.

Note that although the coefficients can be negative, S(n,x) is always a nonnegative number for n,x >= 0.

The degree of S(n,x) is n.

The constant coefficient of S(n,x) is always 0.

Conjecture: the coefficient of x^n in S(n,x) is n*(n+1)/2.

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.

Example

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.

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 *)

Prog

(C++) // see link above

Crossrefs

Cf. A340885 (values of S(n,2)).

Sequence in context: A091921 A037285 A337981 * A181787 A318925 A285213

Adjacent sequences: A341758 A341759 A341760 * A341762 A341763 A341764

Keyword

sign,tabl

Author

Shiyao Guo, Feb 19 2021

Status

approved