Number of Lyndon words of 4 alphabets, One alphabet labeled, the other 3 unlabelled.

Question

1) In the following OEIS Sequence, what is the meaning of One alphabet labeled, the other $3$ unlabeled?

The number of Lyndon words (aperiodic necklaces) with $4n$ beads of $4$ colors, $n$ beads of each color. One color labeled, the other $3$ unlabeled.

$1, 52, 5133, 656880, 97772875, 16032938340, 2812609211657, 518547356184000,\\ 99318297529004400, 19605670296090989500, 3966181169996511862429, 818782296743542722132528, \\171938262068874336023196923,...$

Number of Lyndon words (aperiodic necklaces) with $4n$ beads of 4 colors, $n$ beads of each color.

$1, 6, 312, 30798, 3941280, 586637250, 96197630040, 16875655269942, \\ 3111284137104000, 595909785174026400, 117634021776545937000, 23797087019979071174574,\\ 4912693780461256332795168, 1031629572413246016139181538, 219809927417367517614451764984,...$

2) The first sequence has more conditions, so the first sequence should be smaller than the second one, but the values seems the other way. Where am I making mistake?

Kindly explain me.

Thank you.

Answer 1

Labelled means that the combinatorial objects in question can be distinguished; unlabelled means that they cannot. So $A029809(n) = A074656(n) / 3!$ because A029809 combines Lyndon words of A074656 into equivalence classes under permutations of the three unlabelled colours.

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The first sequence has more conditions, so the first sequence should be smaller than the second one, but the values seems the other way. Where am I making mistake?

The first sequence is smaller than the second one. Perhaps the only mistake you made was not to look at the OFFSET field: A029809 begins 1, 52, 5133 with offset 1, whereas A074656 begins 1, 6, 312, 30798 with offset 0.