A358653 - OEIS

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

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OFFSET

0,3

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

Table of n, a(n) for n=0..14.

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

PROG

(SageMath)

B.<s1, s2> = 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

CROSSREFS

Cf. A354602.

Sequence in context: A359521 A172394 A172395 * A270281 A270728 A270224

Adjacent sequences: A358650 A358651 A358652 * A358654 A358655 A358656

KEYWORD

nonn,more

AUTHOR

Alexei Vernitski, Nov 25 2022

EXTENSIONS

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

STATUS

approved