cp's OEIS Frontend

From Gus Wiseman, Jun 08 2021: (Start)

Also the number of:

- integer partitions of 2n with reverse-alternating sum 2;

- reversed integer partitions of 2n with alternating sum 2;

- integer partitions of 2n with exactly two odd parts, one of which is the greatest;

- odd-length integer partitions of 2n whose conjugate partition has exactly two odd parts.

Note that integer partitions of 2n with alternating or reverse-alternating sum 0 are counted by A000041, ranked by A000290.

(End)

Second binomial transform of A032443 with interpolated zeros.

a(n) is the total number of lattice points, taken over all Dyck n-paths (A000108), that (i) lie on or above ground level and (ii) lie on or directly below a peak. For example with n = 2, UUDD has 1 peak contributing 3 lattice points--(2, 0), (2, 1) and (2, 2) when the path starts at the origin--and UDUD has 2 peaks, each contributing 2 lattice points and so a(2) = 3 + 4 = 7. - David Callan, Jul 14 2006

Hankel transform is binomial(n + 2, 2). - Paul Barry, Dec 04 2007

Image of (-1)^n under the Riordan array ((1/2)(1/(1 - 4x) + 1/sqrt(1 - 4x)), c(x) - 1), c(x) the g.f. of A000108. - Paul Barry, Jun 15 2008

From Gus Wiseman, Jun 21 2021: (Start)

Also the even bisection of A116406 = number compositions of n with alternating sum >= 0, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. The a(3) = 26 compositions are:

(6) (33) (114) (1122) (11112) (111111)

(42) (123) (1131) (11121)

(51) (132) (1221) (11211)

(213) (2112) (12111)

(222) (2121) (21111)

(231) (2211)

(312) (3111)

(321)

(411)

(End)

Also number of partitions of n into exactly 4 parts.

Also the number of weighted cubic graphs on 4 nodes (=the tetrahedron) with weight n. - R. J. Mathar, Nov 03 2018

From Gus Wiseman, Jun 27 2021: (Start)

Also the number of strict integer partitions of 2n with alternating sum 4, or (by conjugation) partitions of 2n covering an initial interval of positive integers with exactly 4 odd parts. The strict partitions with alternating sum 4 are:

(4) (5,1) (6,2) (7,3) (8,4) (9,5) (10,6)

(5,2,1) (5,3,2) (5,4,3) (6,5,3) (7,6,3)

(6,3,1) (6,4,2) (7,5,2) (8,6,2)

(7,4,1) (8,5,1) (9,6,1)

(6,3,2,1) (6,4,3,1) (6,5,4,1)

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

(7,5,3,1)

(8,5,2,1)

(6,4,3,2,1)

(End)

A composition of n is a finite sequence of positive integers summing to n.

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

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.

These partitions are characterized by either being a twin (x,x) or having a wiggly permutation. A sequence is wiggly if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no wiggly permutations, even though it has anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

A variation of problem 11424 in the American Mathematical Monthly. Terms were brute-force calculated using Maple 10.

Proposed Problem 11610 in the Dec 2011 A.M.M.

From Gus Wiseman, Jul 27 2021: (Start)

Also the antidiagonal sums of the matrices counting integer compositions by length and alternating sum (A345197). So a(n) is the number of integer compositions of n + 1 of length (n - s + 3)/2, where s is the alternating sum of the composition. For example, the a(0) = 1 through a(6) = 7 compositions are:

(1) (2) (3) (4) (5) (6) (7)

(11) (21) (31) (41) (51) (61)

(121) (122) (123) (124)

(221) (222) (223)

(1112) (321) (322)

(1211) (1122) (421)

(1221) (1132)

(2112) (1231)

(2211) (2122)

(2221)

(3112)

(3211)

(11131)

(12121)

(13111)

For a bijection with the main (binary string) interpretation, take the run-lengths of each binary string of length n + 1 that satisfies the condition and starts with 1.

(End)

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670. A sequence is alternating (cf. A025047) including pairs (x,x) if there are no adjacent triples (..., x, y, z, ...) where x <= y <= z or x >= y >= z. These sequences avoid the weak consecutive patterns (1,2,3) and (3,2,1).

An alternating pattern of length > 2 is necessarily an anti-run (A005649).

The version without pairs (x,x) is identical to this sequence except a(2) = 2 instead of 3.

In the first Maple program (improvable) AS gives the positive alternating sum of a finite sequence s, EP gives the number of even terms of a finite sequence of positive integers.

For the specified value of n, the second Maple program lists the partitions of n counted by a(n).

Also the number of integer partitions of n with as many even parts as odd parts in the conjugate partition. - Gus Wiseman, Jul 26 2021