cp's OEIS Frontend

A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

An anti-run is a sequence with no adjacent equal parts.

A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.

Alternatively, a multiset is separable if its greatest multiplicity is greater than the sum of its remaining multiplicities plus one.

Also the number of compositions of n whose greatest part is greater than the sum of its remaining parts plus one. For example, the a(2) = 1 through a(7) = 8 compositions are:

(2) (3) (4) (5) (6) (7)

(1,3) (1,4) (1,5) (1,6)

(3,1) (4,1) (2,4) (2,5)

(4,2) (5,2)

(5,1) (6,1)

(1,1,4) (1,1,5)

(1,4,1) (1,5,1)

(4,1,1) (5,1,1)

Also the number of separable strong multisets of length n covering an initial interval of positive integers. A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.

A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.

A multiset is normal if it covers an initial interval of positive integers. The type of a multiset of integers is the unique normal multiset that has the same sequence of multiplicities when its entries are taken in increasing order. For example the type of (3,3,5,5,5,6) is (1,1,2,2,2,3).

A pair h<={g_1,...,g_k} is a combinatory separation iff there exists a multiset partition of h whose multiset of block-types is {g_1,...,g_k}. For example, the (headless) combinatory separations of the multiset (1122) are (1122), (1)(112), (1)(122), (11)(11), (12)(12), (1)(1)(11), (1)(1)(12), (1)(1)(1)(1). This list excludes (12)(11), because one cannot partition (1122) into two blocks where one block has two distinct elements and the other has two equal elements.

We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon factorization of a composition c is a multiset of compositions whose Lyndon product is c.

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

Also the number of multiset partitions of the Lyndon-word factorization of the n-th composition in standard order.

We define the co-Lyndon product of two or more finite sequences to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the co-Lyndon product of (2,3,1) with (2,1,3) is (2,1,2,3,1,3), the product of (2,2,1) with (2,1,3) is (2,1,2,2,1,3), and the product of (1,2,2) with (2,1,2,1) is (1,2,1,2,1,2,2). A co-Lyndon factorization of a composition c is a multiset of compositions whose co-Lyndon product is c.

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

Also the number of multiset partitions of the co-Lyndon-word factorization of the n-th composition in standard order.

An anti-run is a sequence with no adjacent equal parts.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Conjecture: Only vanishes at n = 4 and n = 8.

a(16) = 0. Proof: 16! = 2^15 * m where bigomega(m) = A001222(m) = 13. We can't separate 15 1's with 13 other numbers. - David A. Corneth, Jul 04 2020

We define the co-Lyndon product of two or more finite sequences to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the co-Lyndon product of (2,3,1) with (2,1,3) is (2,1,2,3,1,3), the product of (2,2,1) with (2,1,3) is (2,1,2,2,1,3), and the product of (1,2,2) with (2,1,2,1) is (1,2,1,2,1,2,2). A co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product. Equivalently, a co-Lyndon word is a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into co-Lyndon words, and if these factors are arranged in a certain order, their concatenation is equal to their co-Lyndon product. For example, (1,0,0,1) has co-Lyndon factorization {(1),(1,0,0)}.

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

Also the number of (1,2,1)-matching patterns of length n.

Also the number of (2,1,2)-matching patterns of length n.

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 S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).