cp's OEIS Frontend

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. Then a sequence is (strongly) alternating iff it is a weakly alternating anti-run.

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). An alternating pattern is necessarily an anti-run (A005649).

The version with twins (A344605) is identical to this sequence except with a(2) = 3 instead of 2.

From Gus Wiseman, Jan 16 2022: (Start)

Conjecture: Also the number of weakly up/down patterns of length n, where a sequence is weakly up/down if it is alternately weakly increasing and weakly decreasing, starting with an increase. For example, the a(0) = 1 through a(3) = 6 weakly up/down patterns are:

() (1) (1,1) (1,1,1)

(2,1) (1,1,2)

(2,1,1)

(2,1,2)

(2,1,3)

(3,1,2)

(End)

The number of compatible bipartitions of a set of cardinality n for which at least one subset is not underlined. E.g., for n=2 there are 5 such bipartitions: {1 2}, {1}{2}, {2}{1}, {1}{2}, {2}{1}. A005649 is the number of bipartitions of a set of cardinality n. A000670 is the number of bipartitions of a set of cardinality n with none of the subsets underlined. - Kyle Petersen, Mar 31 2005

a(n) is the cardinality of the image set summed over "all surjections". All surjections means: onto functions f:{1, 2, ..., n} -> {1, 2, ..., k} for every k, 1 <= k <= n. a(n) = Sum_{k=1..n} A019538(n, k)*k. - Geoffrey Critzer, Nov 12 2012

From Gus Wiseman, Jan 15 2022: (Start)

For n > 1, also the number of finite sequences of length n + 1 covering an initial interval of positive integers with at least two adjacent equal parts, or non-anti-run patterns, ranked by the intersection of A348612 and A333217. The complement is counted by A005649. For example, the a(3) = 31 patterns, grouped by sum, are:

(1111) (1222) (1122) (1112) (1233) (1223)

(2122) (1221) (1121) (1332) (1322)

(2212) (2112) (1211) (2133) (2213)

(2221) (2211) (2111) (2331) (2231)

(1123) (3312) (3122)

(1132) (3321) (3221)

(2113)

(2311)

(3112)

(3211)

Also the number of ordered set partitions of {1,...,n + 1} with two successive vertices together in some block.

(End)

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.

This sequence counts compositions that are weakly but not strongly alternating; also weakly alternating non-anti-run compositions.

First differs from 2 * A326291 at a(90) = 4, A326291(90) = 3.

The first odd term is a(144) = 7, whose non-weakly alternating permutations are shown in the example below.

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. Then a sequence is alternating in the sense of A025047 iff it is a weakly alternating anti-run.

For n > 1, the multiset of prime factors of n is row n of A027746. The prime indices A112798 can also be used.

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). An alternating pattern is necessarily an anti-run (A005649).

Conjecture: Also the number of non-weakly up/down (or down/up) patterns of length n. For example:

- The a(3) = 7 non-weakly up/down patterns:

(121), (122), (123), (132), (221), (231), (321)

- The a(3) = 7 non-weakly down/up patterns:

(112), (123), (211), (212), (213), (312), (321)

- The a(3) = 7 non-alternating patterns (see example for more):

(111), (112), (122), (123), (211), (221), (321)

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

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

Also nonsquarefree numbers whose prime factors, taken in order and with multiplicity, are alternately constant and weakly increasing, starting with either.

Also the Heinz numbers of non-strict integer partitions whose part multiplicities, except possibly the first and last, are all even. These are counted by A349795.

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

Also Heinz numbers of non-weakly alternating non-strict integer partitions, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. These partitions are counted by A349796. This sequence involves the somewhat degenerate case where no strict increases are allowed.

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.