cp's OEIS Frontend

Number of linear extensions of the "zig-zag" poset. See ch. 3, prob. 23 of Stanley. - Mitch Harris, Dec 27 2005

Number of increasing 0-1-2 trees on n vertices. - David Callan, Dec 22 2006

Also the number of refinements of partitions. - Heinz-Richard Halder (halder.bichl(AT)t-online.de), Mar 07 2008

The ratio a(n)/n! is also the probability that n numbers x1,x2,...,xn randomly chosen uniformly and independently in [0,1] satisfy x1 > x2 < x3 > x4 < ... xn. - Pietro Majer, Jul 13 2009

For n >= 2, a(n-2) = number of permutations w of an ordered n-set {x_1 < ... x_n} with the following properties: w(1) = x_n, w(n) = x_{n-1}, w(2) > w(n-1), and neither any subword of w, nor its reversal, has the first three properties. The count is unchanged if the third condition is replaced with w(2) < w(n-1). - Jeremy L. Martin, Mar 26 2010

A partition of zigzag permutations of order n+1 by the smallest or the largest, whichever is behind. This partition has the same recurrent relation as increasing 1-2 trees of order n, by induction the bijection follows. - Wenjin Woan, May 06 2011

As can be seen from the asymptotics given in the FORMULA section, one has lim_{n->oo} 2*n*a(n-1)/a(n) = Pi; see A132049/A132050 for the simplified fractions. - M. F. Hasler, Apr 03 2013

a(n+1) is the sum of row n in triangle A008280. - Reinhard Zumkeller, Nov 05 2013

M. Josuat-Verges, J.-C. Novelli and J.-Y. Thibon (2011) give a far-reaching generalization of the bijection between Euler numbers and alternating permutations. - N. J. A. Sloane, Jul 09 2015

Number of treeshelves avoiding pattern T321. Treeshelves are ordered binary (0-1-2) increasing trees where every child is connected to its parent by a left or a right link, see A278678 for more definitions and examples. - Sergey Kirgizov, Dec 24 2016

Number of sequences (e(1), ..., e(n-1)), 0 <= e(i) < i, such that no three terms are equal. [Theorem 7 of Corteel, Martinez, Savage, and Weselcouch] - Eric M. Schmidt, Jul 17 2017

Number of self-dual edge-labeled trees with n vertices under "mind-body" duality. Also number of self-dual rooted edge-labeled trees with n vertices. See my paper linked below. - Nikos Apostolakis, Aug 01 2018

The ratio a(n)/n! is the volume of the convex polyhedron defined as the set of (x_1,...,x_n) in [0,1]^n such that x_i + x_{i+1} <= 1 for every 1 <= i <= n-1; see the solutions by Macdonald and Nelsen to the Amer. Math. Monthly problem referenced below. - Sanjay Ramassamy, Nov 02 2018

Number of total cyclic orders on {0,1,...,n} such that the triple (i-1,i,i+1) is positively oriented for every 1 <= i <= n-1; see my paper on cyclic orders linked below. - Sanjay Ramassamy, Nov 02 2018

The number of binary, rooted, unlabeled histories with n+1 leaves (following the definition of Rosenberg 2006). Also termed Tajima trees, Tajima genealogies, or binary, rooted, unlabeled ranked trees (Palacios et al. 2015). See Disanto & Wiehe (2013) for a proof. - Noah A Rosenberg, Mar 10 2019

From Gus Wiseman, Dec 31 2019: (Start)

Also the number of non-isomorphic balanced reduced multisystems with n + 1 distinct atoms and maximum depth. A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. The labeled version is A006472. For example, non-isomorphic representatives of the a(0) = 1 through a(4) = 5 multisystems are (commas elided):

{1} {12} {{1}{23}} {{{1}}{{2}{34}}} {{{{1}}}{{{2}}{{3}{45}}}}

{{{12}}{{3}{4}}} {{{{1}}}{{{23}}{{4}{5}}}}

{{{{1}{2}}}{{{3}}{{45}}}}

{{{{1}{23}}}{{{4}}{{5}}}}

{{{{12}}}{{{3}}{{4}{5}}}}

Also the number of balanced reduced multisystems with n + 1 equal atoms and maximum depth. This is possibly the meaning of Heinz-Richard Halder's comment (see also A002846, A213427, A265947). The non-maximum-depth version is A318813. For example, the a(0) = 1 through a(4) = 5 multisystems are (commas elided):

{1} {11} {{1}{11}} {{{1}}{{1}{11}}} {{{{1}}}{{{1}}{{1}{11}}}}

{{{11}}{{1}{1}}} {{{{1}}}{{{11}}{{1}{1}}}}

{{{{1}{1}}}{{{1}}{{11}}}}

{{{{1}{11}}}{{{1}}{{1}}}}

{{{{11}}}{{{1}}{{1}{1}}}}

(End)

With s_n denoting the sum of n independent uniformly random numbers chosen from [-1/2,1/2], the probability that the closest integer to s_n is even is exactly 1/2 + a(n)/(2*n!). (See Hambardzumyan et al. 2023, Appendix B.) - Suhail Sherif, Mar 31 2024

The number of permutations of size n+1 that require exactly n passes through a stack (i.e. have reverse-tier n-1) with an algorithm that prioritizes outputting the maximum possible prefix of the identity in a given pass and reverses the remainder of the permutation for prior to the next pass. - Rebecca Smith, Jun 05 2024

For n > 1, a(n) is the expected number of tosses of a fair coin to get n-1 consecutive heads. - Pratik Poddar, Feb 04 2011

For n > 2, Sum_{k=1..a(n)} (-1)^binomial(n, k) = A064405(a(n)) + 1 = 0. - Benoit Cloitre, Oct 18 2002

For n > 0, the number of nonempty proper subsets of an n-element set. - Ross La Haye, Feb 07 2004

Numbers j such that abs( Sum_{k=0..j} (-1)^binomial(j, k)*binomial(j + k, j - k) ) = 1. - Benoit Cloitre, Jul 03 2004

For n > 2 this formula also counts edge-rooted forests in a cycle of length n. - Woong Kook (andrewk(AT)math.uri.edu), Sep 08 2004

For n >= 1, conjectured to be the number of integers from 0 to (10^n)-1 that lack 0, 1, 2, 3, 4, 5, 6 and 7 as a digit. - Alexandre Wajnberg, Apr 25 2005

Beginning with a(2) = 2, these are the partial sums of the subsequence of A000079 = 2^n beginning with A000079(1) = 2. Hence for n >= 2 a(n) is the smallest possible sum of exactly one prime, one semiprime, one triprime, ... and one product of exactly n-1 primes. A060389 (partial sums of the primorials, A002110, beginning with A002110(1)=2) is the analog when all the almost primes must also be squarefree. - Rick L. Shepherd, May 20 2005

From the second term on (n >= 1), the binary representation of these numbers is a 0 preceded by (n - 1) 1's. This pattern (0)111...1110 is the "opposite" of the binary 2^n+1: 1000...0001 (cf. A000051). - Alexandre Wajnberg, May 31 2005

The numbers 2^n - 2 (n >= 2) give the positions of 0's in A110146. Also numbers k such that k^(k + 1) = 0 mod (k + 2). - Zak Seidov, Feb 20 2006

Partial sums of A155559. - Zerinvary Lajos, Oct 03 2007

Number of surjections from an n-element set onto a two-element set, with n >= 2. - Mohamed Bouhamida, Dec 15 2007

It appears that these are the numbers n such that 3*A135013(n) = n*(n + 1), thus answering Problem 2 on the Mathematical Olympiad Foundation of Japan, Final Round Problems, Feb 11 1993 (see link Japanese Mathematical Olympiad).

Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if x is a proper subset of y or y is a proper subset of x and x and y are disjoint. Then a(n+1) = |R|. - Ross La Haye, Mar 19 2009

The permutohedron Pi_n has 2^n - 2 facets [Pashkovich]. - Jonathan Vos Post, Dec 17 2009

First differences of A005803. - Reinhard Zumkeller, Oct 12 2011

For n >= 1, a(n + 1) is the smallest even number with bit sum n. Cf. A069532. - Jason Kimberley, Nov 01 2011

a(n) is the number of branches of a complete binary tree of n levels. - Denis Lorrain, Dec 16 2011

For n>=1, a(n) is the number of length-n words on alphabet {1,2,3} such that the gap(w)=1. For a word w the gap g(w) is the number of parts missing between the minimal and maximal elements of w. Generally for words on alphabet {1,2,...,m} with g(w)=g>0 the e.g.f. is Sum_{k=g+2..m} (m - k + 1)*binomial((k - 2),g)*(exp(x) - 1)^(k - g). a(3)=6 because we have: 113, 131, 133, 311, 313, 331. Cf. A240506. See the Heubach/Mansour reference. - Geoffrey Critzer, Apr 13 2014

For n > 0, a(n) is the minimal number of internal nodes of a red-black tree of height 2*n-2. See the Oct 02 2015 comment under A027383. - Herbert Eberle, Oct 02 2015

Conjecture: For n>0, a(n) is the least m such that A007814(A000108(m)) = n-1. - L. Edson Jeffery, Nov 27 2015

Actually this follows from the procedure for determining the multiplicity of prime p in C(n) given in A000108 by Franklin T. Adams-Watters: For p=2, the multiplicity is the number of 1 digits minus 1 in the binary representation of n+1. Obviously, the smallest k achieving "number of 1 digits" = k is 2^k-1. Therefore C(2^k-2) is divisible by 2^(k-1) for k > 0 and there is no smaller m for which 2^(k-1) divides C(m) proving the conjecture. - Peter Schorn, Feb 16 2020

For n >= 0, a(n) is the largest number you can write in bijective base-2 (a.k.a. the dyadic system, A007931) with n digits. - Harald Korneliussen, May 18 2019

The terms of this sequence are also the sum of the terms in each row of Pascal's triangle other than the ones. - Harvey P. Dale, Apr 19 2020

For n > 1, binomial(a(n),k) is odd if and only if k is even. - Charlie Marion, Dec 22 2020

For n >= 2, a(n+1) is the number of n X n arrays of 0's and 1's with every 2 X 2 square having density exactly 2. - David desJardins, Oct 27 2022

For n >= 1, a(n+1) is the number of roots of unity in the unique degree-n unramified extension of the 2-adic field Q_2. Note that for each p, the unique degree-n unramified extension of Q_p is the splitting field of x^(p^n) - x, hence containing p^n - 1 roots of unity for p > 2 and 2*(2^n - 1) for p = 2. - Jianing Song, Nov 08 2022

The permutation 732569148 has 4 alternating runs: 732, 2569, 91 and 148.

a(n) mod 10 for n >= 2 is the periodic sequence repeat: 0, 1, 6, 9.

For n >= 2, a(n) is the number of permutations on [n] that have n-2 "sequences" (which are maximal monotone runs in Comtet terminology) and start increasing. - Michael Somos, Aug 28 2013

From Petros Hadjicostas, Aug 07 2019: (Start)

With regard to the comment by Michael Somos above, a "sequence" in a permutation according to Ex. 13 (pp. 260-261) of Comtet (1974) is actually a "séquence" in a permutation according to André. He uses this terminology in several of his papers cited in the links below.

In the terminology of array A059427, these so-called "séquences" in permutations defined by Comtet and André are called "alternate runs" (or just "runs"). We discuss these so-called "sequences" below.

We clarify that a(n) is actually one-half the number of permutations on [n] that have n-2 "séquences" as defined by Comtet and André.

André (1884) defines a "maximum" of a permutation of [n] to be any number in the permutation that is greater than both of its neighbors, if it is an interior number, or greater than its single neighbor, if it is either at the beginning or at the end of the permutation.

André (1884) also defines a "minimum" of a permutation of [n] to be any number in the permutation that is less than both of its neighbors, if it is an interior number, or less than its single neighbor, if it is either at the beginning or at the end of the permutation.

A "séquence" in a permutation of [n] according to André and Comtet is a list of consecutive numbers in the permutation that begins with a maximum and ends with a minimum, or vice versa, but has no interior maxima and minima. As mentioned above, other authors call these so-called "séquences" "alternate runs" (or just "runs").

For example, in the permutation 78125436 of [8], we have three maxima, 8, 5, and 6; three minima, 7, 1, and 3; and the so-called "séquences" ("alternate runs") 78, 81, 125, 543, 36 (see p. 122 in André (1884)).

If in the above permutation we take the difference 8-7, 1-8, 2-1, 5-2, 4-5, 3-4, 6-3, we may form a word (list) of signs of consecutive differences: +-++--+.

In general, if in a permutation of [n], say a_1,a_2,...,a_n (written in one-line notation, but not in cycle notation), we form the differences a_2 - a_1, a_3 - a_2, ..., a_n - a_{n-1}, then we get a list of n-1 signs (+ or -).

For n >= 2, André (1885) calls a permutation of [n] "alternate" if it has n - 1 so-called "séquences" ("alternate runs"); i.e., if the corresponding list of signs alternates between + and -. See the documentation and references for sequences A000111 and A001250.

For n >= 2, André (1885) calls a permutation of [n] "quasi-alternate" if it has n - 2 so-called "séquences" ("alternate runs"); i.e., if the corresponding list of signs alternates between + and - except for a single ++ or a single --, but not both.

In the above example, the permutation 78125436 has 5 so-called "séquences" ("alternate signs") and 5 < 8-2 < 8-1; that is, it is neither alternate nor quasi-alternate. We can reach the same conclusion by looking at its corresponding list of signs +-++--+. The permutation is neither alternate nor quasi-alternate because we have one ++ and one --.

On p. 316, André (1885) gives the following two examples of permutations of [8]: 31426587 and 32541768 (using one-line notation for permutations). The first one has list of signs -+-+-+- while the second one has list of signs -+--+-+. The first one is alternate while the second one is quasi-alternate (because of a single --). Alternatively, the first one has n-1 = 7 so-called "séquences" ("alternate runs")--31, 14, 42, 26, 65, 58, 87--while the second one has n-2 = 6 so-called "séquences" ("alternate runs")--32, 25, 541, 17, 76, 68.

Here 2*a(n) is the total number of quasi-alternate permutations of [n]. Actually, André (1884, 1885) uses P_{n,s} to denote the number of permutations of [n] with exactly s of his so-called "séquences" ("alternate runs"). He uses the notation A_n to denote half the number of alternate permutations of [n] and B_n to denote half the number of quasi-alternate permutations of [n].

Thus, P_{n,n-1} = 2*A_n = 2*A000111(n) = A001250(n) for n >= 2 and P_{n,n-2} = 2*B_n = 2*a(n) for n >= 2.

We have P_{n,s} = A059427(n,s) for n >= 2 and s >= 1. See also p. 261 in Comtet (1974). They satisfy André's recurrence P_{n,s} = s*P_{n-1,s} + 2*P_{n-1,s-1} + (n-s)*P_{n-1,s-2} for n >= 3 and s >= 3 with P(n, 1) = 2 for n >= 2 and P(n, s) = 0 for s >= n.

The numbers Q(n,s) that count the circular permutations of [n] with exactly s so-called "séquences" ("alternate runs") appear in array A008303. They have also been studied by Désiré André (see the references there).

(End)

Table starts

.......1........1.........1.........1.........1.........1.........1.........1

.......2........2.........2.........2.........2.........2.........2.........2

.......4........6.........6.........6.........6.........6.........6.........6

......10.......22........24........24........24........24........24........24

......32.......90.......118.......120.......120.......120.......120.......120

.....122......422.......658.......718.......720.......720.......720.......720

.....544.....2226......4078......4914......5038......5040......5040......5040

....2770....13102.....27724.....37300.....40066.....40318.....40320.....40320

...15872....85170....205134....308460....353556....362370....362878....362880

..101042...606542...1641534...2748354...3399246...3600306...3627778...3628798

..707584..4697946..14132390..26194542..35142546..38963958..39830282..39914754

.5405530.39330982.130299584.265691456.387129588.453658380.475089392.478739984

Row n has 1+floor(n/2) terms.

T(n,k) is also the number of permutations of [n] with k "exterior peaks" where we count peaks in the usual way, but add a peak at the beginning if the permutation begins with a descent, e.g. 213 has one exterior peak. T(3,1) = 5 since all permutations of [3] have an exterior peak except 123. This is also the definition for peaks of signed permutations, where we assume that a signed permutation always begins with a zero. - Kyle Petersen, Jan 14 2005

From Petros Hadjicostas, Aug 09 2019: (Start)

In their book, David and Barton (1962) use the notation T_{N,v*}^* for this array, where N is the length of the permutation and v* is the so-called "number of runs up" in the permutation. In modern terminology, a "run up" in a permutation is an increasing run of length >= 2. See their discussion on p. 154 of their book and see p. 163 for the definition of T_{N,v*}^*.

They do not consider as "runs up" single elements b_i in a permutation b = (b_1, b_2, ..., b_n) even if they satisfy b_{i-1} > b_i > b_{i+1} (with b_{n-1} > b_n when i = n and b_1 > b_2 when i = 1). (The command Runs[b] for permutation b in Mathematica, using the package Combinatorica`, will generate not only the "runs up" of b but also the single elements in the permutation b that satisfy one of the above inequalities. For example, Runs[{3,2,1}] gives the set of runs {{3}, {2}, {1}}, none of which is a "run up".)

So, here n = N and k = v*. On p. 163 of their book they give the recurrence shown below in the FORMULA section from Charalambides' (2002) book: T(n+1, k) = (2*k + 1) * T(n,k) + (n - 2*k + 2) * T(n, k-1) for n >= 0 and 1 <= k <= floor(n/2) + 1. The values of T_{N,v*}^* = T(n, k) appear in Table 10.5 (p. 163) of their book.

Since the complement of a permutation (b_1, b_2, ..., b_n) is (n+1-b_1, n+1-b_2, ..., n+1-b_n), it is clear that the current array T(n, k) is also the number of permutations of [n] with k decreasing runs of length >= 2 (i.e., the number of permutations of [n] with k "runs down" according to David and Barton (1962)).

Note that the number of permutations of [n] with k runs of length >= 2 that are either increasing or decreasing (i.e., the number of permutations of [n] that contain k "runs up" and "runs down" in total) is given by array A059427. One half of array A059427 is given in Table 10.4 (p. 159) in David and Barton (1962)--see also array A008970.

A run that is either a "run up" or "run down" (i.e., an ascending or a descending run of length >= 2) is called "séquence" by André (19th century) and Comtet (1974). See the references for sequence A000708 or for array A059427. (Again, recall that David and Barton do not consider single numbers as either a "run up" or a "run down".)

Morley (1897) proved that in a permutation of [n], #("runs up") + #("runs down") + #(monotonic triplets of adjacent numbers in the permutation) = n - 1. (His definition of a run is highly non-standard!) See the example below.

The number Q(n,k) of circular permutations of [n] that contain k runs that are either "runs up" or "runs down" (that is, k ascending or descending runs of length >= 2) is given by array A008303. More precisely, Q(n+1, 2*(k+1)) = A008303(n, k) for n >= 1 and 0 <= k <= ceiling(n/2)-1. In addition, Q(n, s) = 0 when either s is odd, or n <= 1, or s > n. Also, Q_{n,2} = 2^(n-2) for n >= 2.

The numbers in array A008303 appear in Table 10.6 (p. 163) in David and Barton (1962).

(End)

The number of permutations of [n] with n-2 sequences (see Comtet).

From Petros Hadjicostas, Aug 08 2019: (Start)

We clarify the word "sequences" used above because it may not be standard. On pp. 260-261 of his book, Comtet (1974) defines a so-called "sequence" in a permutation b of [n]. Using one-line notation (not cycle notation), write b = (b_1, b_2, ..., b_n) for the elements of a permutation of [n]. A maximal list of indices of length l (where l >= 2) is called a "sequence" in the permutation b if it is of the form {i, i+1, ..., i+l-1} for some integer i (with 1 <= i <= n-l+1) such that b_i < b_{i+1} < ... < b_{i+l-1} or b_i > b_{i+1} > ... > b_{i+l-1}. (The word "maximal" means that in the first case, b_{i-1} > b_i and b_{i+l} < b_{i+l-1}, while in the second case, b_{i-1} < b_i and b_{i+l} > b_{i+l-1}, provided that b_{i-1} and b_{i+l} can be defined.) The assumption l >= 2 is important; i.e., these so-called "sequences" should have length >= 2.

Comtet (1974) has borrowed this confusing terminology about "séquences" in permutations from André (see links to some of his papers below). André actually uses the term "séquence" for the list of terms (b_i, b_{i+1}, ..., b_{i+l-1}) rather than the index set {i, i+1, ..., i+l-1}.

Some authors today use the term "alternate runs" (or just "runs") to discuss these so-called "séquences" defined by Comtet and André but we must have l >= 2.

Thus, here a(n) is the number of permutations of [n] with n-2 such "séquences" ("alternate runs").

For an extensive discussion of these so-called "séquences" in permutations ("alternate runs"), maxima and minima in a permutation, alternate and quasi-alternate permutations, and other related information, see the four papers by André, or see my comments for sequence A000708 (which equals one-half of the current sequence).

David and Barton (1962, p. 154) call these "séquences" "runs up" if they are ascending and "runs down" if they are descending. In modern terminology, "runs up" are ascending runs of length >= 2 while "runs down" are descending runs of length >= 2. Thus, a modern terminology for these "séquences" is "ascending or descending runs of length >= 2".

(End)