A358653 a(n) is the number of trivial braids on 3 strands which are products of n generators a, b, where a = sigma_1 sigma_2 sigma_1 and b = sigma_1 sigma_2.

1, 0, 4, 0, 28, 10, 244, 210, 2412, 3366, 26014, 49456, 299452, 701818, 3624478

Offset: 0

Author

Alexei Vernitski, Nov 25 2022

Keywords

• nonn
• more

Comments

In the discussion of A354602, Andrey Zabolotskiy asked what the values of the sequence would be if expressed in terms of a and b. This sequence lists these values.

Links

• Wikipedia, Relation with symmetric group and the pure braid group.

Crossrefs

Cf. A354602.

Programs

SageMath

B. = BraidGroup(3)
gen = [s1*s2*s1, s1*s2]
gen += [x^-1 for x in gen]
e = B(())
words, a = {e: 1}, [1]
for n in range(15):
    old_words, words = words, {}
    for w, c in old_words.items():
        for g in gen:
            nw = w*g
            words[nw] = words.get(nw, 0) + c
    a.append(words.get(e, 0))
print(a) # Andrey Zabolotskiy, Jan 16 2024

Extensions

a(11)-a(14) from Andrey Zabolotskiy, Jan 16 2024