cp's OEIS Frontend

Area under Dyck excursions (paths ending in 0): a(n) is the sum of the areas under all Dyck excursions of length 2*n (nonnegative walks beginning and ending in 0 with jumps -1,+1).

Number of inversions in all 321-avoiding permutations of [n+1]. Example: a(2)=6 because the 321-avoiding permutations of [3], namely 123,132,312,213,231, have 0, 1, 2, 1, 2 inversions, respectively. - Emeric Deutsch, Jul 28 2003

Convolution of A001791 and A000984. - Paul Barry, Feb 16 2005

a(n) = total semilength of "longest Dyck subpath" starting at an upstep U taken over all upsteps in all Dyck paths of semilength n. - David Callan, Jul 25 2008

[1,6,29,130,562,2380,...] is convolution of A001700 with itself. - Philippe Deléham, May 19 2009

From Ran Pan, Feb 04 2016: (Start)

a(n) is the total number of times that all the North-East lattice paths from (0,0) to (n+1,n+1) bounce off the diagonal y = x to the right. This is related to paired pattern P_2 in Pan and Remmel's link and more details can be found in Section 3.2 in the link.

a(n) is the total number of times that all the North-East lattice paths from (0,0) to (n+1,n+1) horizontally cross the diagonal y = x. This is related to paired pattern P_3 in Pan and Remmel's link and more details can be found in Section 3.3 in the link.

2*a(n) is the total number of times that all the North-East lattice paths from (0,0) to (n+1,n+1) bounce off the diagonal y = x. This is related to paired pattern P_2 and P_4 in Pan and Remmel's link and more details can be found in Section 4.2 in the link.

2*a(n) is the total number of times that all the North-East lattice paths from (0,0) to (n+1,n+1) cross the diagonal y = x. This is related to paired pattern P_3 and P_4 in Pan and Remmel's link and more details can be found in Section 4.3 in the link. (End)

From Gus Wiseman, Jul 17 2021: (Start)

Also the number of integer compositions of 2*(n+1) with alternating sum < 0, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. For example, the a(3) = 29 compositions of 8 are:

(1,7) (1,5,2) (1,1,1,5) (1,1,1,4,1) (1,1,1,1,1,3)

(2,6) (1,6,1) (1,1,2,4) (1,2,1,3,1) (1,1,1,2,1,2)

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

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

(1,3,1,3) (1,2,1,2,1,1)

(1,3,2,2) (1,3,1,1,1,1)

(1,4,1,2)

(1,4,2,1)

(1,5,1,1)

(2,1,1,4)

(2,2,1,3)

(2,3,1,2)

(2,4,1,1)

Also the number of integer compositions of 2*(n+1) with reverse-alternating sum < 0. For a bijection, keep the odd-length compositions and reverse the even-length ones.

Also the number of 2*(n+1)-digit binary numbers with more 0's than 1's. For example, the a(2) = 6 binary numbers are: 100000, 100001, 100010, 100100, 101000, 110000; or in decimal: 32, 33, 34, 36, 40, 48.

(End)

Number of subsets of {1,2,...,n} that contain more even than odd numbers.

Note that A058622 counts the nonempty subsets of {1,2,...,n} that contain more odd than even numbers.

From Gus Wiseman, Jul 22 2021: (Start)

Also the number of integer compositions of n + 1 with alternating sum < 0, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. For example, the a(0) = 0 through a(6) = 6 compositions (empty columns indicated by dots) are:

. . (12) (13) (14) (15)

(23) (24)

(131) (141)

(1112) (1113)

(1211) (1212)

(1311)

Also the number of integer compositions of n + 1 with reverse-alternating sum < 0. For a bijection, keep the odd-length compositions and reverse the even-length ones.

Also the number of (n+1)-digit binary numbers with more 0's than 1's. For example, the a(0) = 0 through a(5) = 6 binary numbers are:

. . 100 1000 10000 100000

10001 100001

10010 100010

10100 100100

11000 101000

110000

(End)

2*a(n) is the number of all-positive pinnacle sets that are admissible in the group S_{n+1}^B of signed permutations, but not admissible in S_{n+1}. - Bridget Tenner, Jan 06 2023

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

Also the number of reversed of integer partitions of n with alternating sum < 0.

No integer partitions have alternating sum < 0, so the non-reversed version is all zeros.

Is this sequence weakly increasing? Note: a(2n + 2) = A236914(n), a(2n) = A344743(n).

A formula for the reverse-alternating sum of a partition is: (-1)^(k-1) times the number of odd parts in the conjugate partition, where k is the number of parts. So a(n) is the number of integer partitions of n of even length whose conjugate parts are not all odd. Partitions of the latter type are counted by A086543. By conjugation, a(n) is also the number of integer partitions of n of even maximum whose parts are not all odd.

The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

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.

The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

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.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

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.

The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

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.

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.

The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. Of course, the reverse-alternating sum of prime indices is also the alternating sum of reversed prime indices.

Also numbers with exactly one odd conjugate prime index. Conjugate prime indices are listed by A321650, ranked by A122111.

The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

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.

All such factorizations have even length and alternating sum < 0, so partitions of this type are counted by A344608.

Also the number of factorizations of n with alternating sum < 0.

A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).