cp's OEIS Frontend

Number of permutations of degree n with at most one fall; called "Grassmannian permutations" by Lascoux and Schützenberger. - Axel Kohnert (Axel.Kohnert(AT)uni-bayreuth.de)

Number of different permutations of a deck of n cards that can be produced by a single shuffle. [DeSario]

Number of Dyck paths of semilength n having at most one long ascent (i.e., ascent of length at least two). Example: a(4)=12 because among the 14 Dyck paths of semilength 4, the only paths that have more than one long ascent are UUDDUUDD and UUDUUDDD (each with two long ascents). Here U = (1, 1) and D = (1, -1). Also number of ordered trees with n edges having at most one branch node (i.e., vertex of outdegree at least two). - Emeric Deutsch, Feb 22 2004

Number of {12,1*2*,21*}-avoiding signed permutations in the hyperoctahedral group.

Number of 1342-avoiding circular permutations on [n+1].

2^n - n is the number of ways to partition {1, 2, ..., n} into arithmetic progressions, where in each partition all the progressions have the same common difference and have lengths at least 1. - Marty Getz (ffmpg1(AT)uaf.edu) and Dixon Jones (fndjj(AT)uaf.edu), May 21 2005

if b(0) = x and b(n) = b(n-1) + b(n-1)^2*x^(n-2) for n > 0, then b(n) is a polynomial of degree a(n). - Michael Somos, Nov 04 2006

The chromatic invariant of the Mobius ladder graph M_n for n >= 2. - Jonathan Vos Post, Aug 29 2008

Dimension sequence of the dual alternative operad (i.e., associative and satisfying the identity xyz + yxz + zxy + xzy + yzx + zyx = 0) over the field of characteristic 3. - Pasha Zusmanovich, Jun 09 2009

An elephant sequence, see A175654. For the corner squares six A[5] vectors, with decimal values between 26 and 176, lead to this sequence (without the first leading 1). For the central square these vectors lead to the companion sequence A168604. - Johannes W. Meijer, Aug 15 2010

a(n+1) is also the number of order-preserving and order-decreasing partial isometries (of an n-chain). - Abdullahi Umar, Jan 13 2011

A040001(n) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n. - Michael Somos, May 12 2012

A130103(n+1) = p(n+1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n. - Michael Somos, May 12 2012

The number of labeled graphs with n vertices whose vertex set can be partitioned into a clique and a set of isolated points. - Alex J. Best, Nov 20 2012

For n > 0, a(n) is a B_2 sequence. - Thomas Ordowski, Sep 23 2014

See coefficients of the linear terms of the polynomials of the table on p. 10 of the Getzler link. - Tom Copeland, Mar 24 2016

Consider n points lying on a circle, then for n>=2 a(n-1) is the maximum number of ways to connect two points with non-intersecting chords. - Anton Zakharov, Dec 31 2016

Also the number of cliques in the (n-1)-triangular honeycomb rook graph. - Eric W. Weisstein, Jul 14 2017

From Eric M. Schmidt, Jul 17 2017: (Start)

Number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) != e(j) < e(k). [Martinez and Savage, 2.7]

Number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i), e(j), e(k) pairwise distinct. [Martinez and Savage, 2.7]

Number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(j) >= e(k) and e(i) != e(k) pairwise distinct. [Martinez and Savage, 2.7]

(End)

Number of F-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are F-equivalent iff the positions of pattern F are identical in these paths. - Sergey Kirgizov, Apr 08 2018

From Gus Wiseman, Feb 10 2019: (Start)

Also the number of connected partitions of an n-cycle. For example, the a(1) = 1 through a(4) = 12 connected partitions are:

{{1}} {{12}} {{123}} {{1234}}

{{1}{2}} {{1}{23}} {{1}{234}}

{{12}{3}} {{12}{34}}

{{13}{2}} {{123}{4}}

{{1}{2}{3}} {{124}{3}}

{{134}{2}}

{{14}{23}}

{{1}{2}{34}}

{{1}{23}{4}}

{{12}{3}{4}}

{{14}{2}{3}}

{{1}{2}{3}{4}}

(End)

Number of subsets of n-set without the single-element subsets. - Yuchun Ji, Jul 16 2019

For every prime p, there are infinitely many terms of this sequence that are divisible by p (see IMO Compendium link and Doob reference). Corresponding indices n are: for p = 2, even numbers A299174; for p = 3, A047257; for p = 5, A349767. - Bernard Schott, Dec 10 2021

Primes are in A081296 and corresponding indices in A048744. - Bernard Schott, Dec 12 2021

Square array read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

The topmost row (row 1) of the array is A000079 (powers of 2), and in general each row 2^k contains the sequence (2^n - k), starting from the term (2^(k+1) - k). This follows from the properties (3) and (4) of A004001 given on page 227 of Kubo & Vakil paper (page 3 in PDF).

Moreover, each row 2^k - 1 (for k >= 2) contains the sequence 2^n - n - (k-2), starting from the term (2^(k+1) - (2k-1)). To see why this holds, consider the definitions of sequences A162598 and A265332, the latter which also illustrates how the frequency counts Q_n for A004001 are recursively constructed (in the Kubo & Vakil paper).

Without its first term, it is the binomial transform of 1,1,1,1,2,2,2,2,2...

Partial sums are A084637 (without leading 1).

The sequence starting 1,2,4,... is the binomial transform of (1,1,1,1,1,2,2,2,...) with b(n) = Sum_{k=0..4} C(n,k) + 2*Sum_{k=5..n} C(n,k) = 2^(n+1) - (n^4 -2*n^3 + 11*n^2 + 14*n + 24)/24. This gives the partial sums of A084635.

The sequence starting 1,2,4,... is the binomial transform of (1, 1, 1, 1, 1, 1, 2, 2, 2, ...) with A035038(n) = Sum_{k=0..5} C(n,k) + 2*Sum_{k=6..n} C(n,k) = 2^n - (n^5 - 5*n^4 + 25*n^3 + 5*n^2 + 94*n + 120)/120. This gives the partial sums of A084636.

The sequence starting 1,2,4,... is the binomial transform of (1,1,1,1,1,1,1,2,2...) with a(n) = Sum_{k=0..6} C(n,k) + 2*Sum_{k=7..n} C(n,k) = 2^(n+1) - A008859(n). This gives the partial sums of A084637.