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stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;

wigQ[y_]:=Or[Length[y]==0, Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];

A025047 counts alternating or wiggly compositions, directed also A025048, A025049.

A345164 counts alternating permutations of prime indices (with twins: A344606).

A345165 counts partitions without w/o an alternating permutation, ranked by A345171.

A345170 counts partitions with w/ an alternating permutation, ranked by A345172.

Alternating or wiggly compositions are counted by A025047, directed A025048, A025049.

Non-alternating compositions are counted by A345192.

A025047 counts alternating or wiggly compositions, directed A025048, A025049.

A345165 counts partitions without a an alternating permutation, ranked by A345171.

A345170 counts partitions with a an alternating permutation, ranked by A345172.

A345192 counts non-alternating compositions.

Numbers k such that the k-th composition in standard order is a non-wiggly alternating anti-run.

A sequence is wiggly alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no wiggly alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

Wikipedia, <a href="https://en.wikipedia.org/wiki/Alternating_permutation">Alternating permutation</a>

Intersection of A345168 (non-wigglyalternating) and A333489 (anti-run).

Wiggly Alternating or wiggly compositions are counted by A025047 (ascend: , directed A025048, descend: A025049).

Non-wiggly alternating compositions are counted by A345192.

A001250 counts wiggly alternating permutations, complement A348615.

A345164 counts wiggly alternating permutations of prime indices (with twins: A344606).

A345165 counts partitions without a wiggly alternating permutation, ranked by A345171.

A345170 counts partitions with a wiggly alternating permutation, ranked by A345172.

A345194 counts wiggly alternating patterns (with twins: A344605).

- Non-anti-runs are A348612.

- Wiggly Alternating compositions are A345167.

- Non-wiggly Alternating compositions are A345168.

Cf. A001222, A008965, A238279, A344614, A344615, A344652, A344653, A344654, A345162, A345163, A345193, A348609, A348613.

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Cf. A001222, A008965, A238279, `A344612, A344614, A344615, `A344652, A344653, A344654, A345162, A345163, A345193.

A344654 counts non-twin partitions without a wiggly permutation, ranked by A344653.

Cf. A001222, A008965, A238279, `A344612, A344614, A344615, `A344652, A344653, A344654, A345162, A345163, A345193.