A008965 -id:A008965 - cp's OEIS frontend

A345168 Numbers k such that the k-th composition in standard order is not alternating.

3, 7, 10, 11, 14, 15, 19, 21, 23, 26, 27, 28, 29, 30, 31, 35, 36, 37, 39, 42, 43, 46, 47, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 69, 71, 73, 74, 75, 78, 79, 83, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 99, 100, 101, 103, 104, 105, 106, 107, 110

Offset: 1

Gus Wiseman, Jun 15 2021

nonn

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 sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

			The sequence of terms together with their binary indices begins:
     3: (1,1)          35: (4,1,1)        59: (1,1,2,1,1)
     7: (1,1,1)        36: (3,3)          60: (1,1,1,3)
    10: (2,2)          37: (3,2,1)        61: (1,1,1,2,1)
    11: (2,1,1)        39: (3,1,1,1)      62: (1,1,1,1,2)
    14: (1,1,2)        42: (2,2,2)        63: (1,1,1,1,1,1)
    15: (1,1,1,1)      43: (2,2,1,1)      67: (5,1,1)
    19: (3,1,1)        46: (2,1,1,2)      69: (4,2,1)
    21: (2,2,1)        47: (2,1,1,1,1)    71: (4,1,1,1)
    23: (2,1,1,1)      51: (1,3,1,1)      73: (3,3,1)
    26: (1,2,2)        52: (1,2,3)        74: (3,2,2)
    27: (1,2,1,1)      53: (1,2,2,1)      75: (3,2,1,1)
    28: (1,1,3)        55: (1,2,1,1,1)    78: (3,1,1,2)
    29: (1,1,2,1)      56: (1,1,4)        79: (3,1,1,1,1)
    30: (1,1,1,2)      57: (1,1,3,1)      83: (2,3,1,1)
    31: (1,1,1,1,1)    58: (1,1,2,2)      84: (2,2,3)

Wikipedia, Alternating permutation

The complement is A345167.
These compositions are counted by A345192.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A025047 counts alternating or wiggly compositions, directed A025048, A025049.
A344604 counts alternating compositions with twins.
A345194 counts alternating patterns (with twins: A344605).
A345164 counts alternating permutations of prime indices (with twins: A344606).
A345165 counts partitions without a alternating permutation, ranked by A345171.
A345170 counts partitions with a alternating permutation, ranked by A345172.
A348610 counts alternating ordered factorizations, complement A348613.
Statistics of standard compositions:
- Length is A000120.
- Constant runs are A124767.
- Heinz number is A333219.
- Number of maximal anti-runs is A333381.
- Runs-resistance is A333628.
- Number of distinct parts is A334028.
Classes of standard compositions:
- Weakly decreasing compositions (partitions) are A114994.
- Weakly increasing compositions (multisets) are A225620.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Anti-run compositions are A333489.
- Non-anti-run compositions are A348612.
Cf. A001222, A005649, A008965, A059893, A106356, A238279, A344615, A345162, A345163, A345166, A345169, A348377, A348380.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Select[Range[0,100],Not@*wigQ@*stc]

A351013 Number of integer compositions of n with all distinct runs.

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 26, 48, 88, 161, 294, 512, 970, 1634, 2954, 5156, 9119, 15618, 27354, 46674, 80130, 138078, 232286, 394966, 665552, 1123231, 1869714, 3146410, 5186556, 8620936, 14324366, 23529274, 38564554, 63246744, 103578914, 167860584, 274465845

Offset: 0

Gus Wiseman, Feb 09 2022

nonn

			The a(1) = 1 through a(5) = 14 compositions:
  (1)  (2)    (3)      (4)        (5)
       (1,1)  (1,2)    (1,3)      (1,4)
              (2,1)    (2,2)      (2,3)
              (1,1,1)  (3,1)      (3,2)
                       (1,1,2)    (4,1)
                       (2,1,1)    (1,1,3)
                       (1,1,1,1)  (1,2,2)
                                  (2,2,1)
                                  (3,1,1)
                                  (1,1,1,2)
                                  (1,1,2,1)
                                  (1,2,1,1)
                                  (2,1,1,1)
                                  (1,1,1,1,1)
For example, the composition c = (3,1,1,1,1,2,1,1,3,4,1,1) has runs (3), (1,1,1,1), (2), (1,1), (3), (4), (1,1), and since (3) and (1,1) both appear twice, c is not counted under a(20).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The version for run-lengths instead of runs is A329739, normal A329740.
These compositions are ranked by A351290, complement A351291.
A000005 counts constant compositions, ranked by A272919.
A005811 counts runs in binary expansion.
A011782 counts integer compositions.
A059966 counts binary Lyndon compositions, necklaces A008965, aperiodic A000740.
A116608 counts compositions by number of distinct parts.
A238130 and A238279 count compositions by number of runs.
A242882 counts compositions with distinct multiplicities.
A297770 counts distinct runs in binary expansion.
A325545 counts compositions with distinct differences.
A329744 counts compositions by runs-resistance.
A351014 counts distinct runs in standard compositions.
Counting words with all distinct runs:
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Cf. A003242, A025047, A044813, A098504, A098859, A106356, A329738, A328592, A334028, A351015, A351201, A351204.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Split[#]&]],{n,0,10}]

\\ here LahI is A111596 as row polynomials.
LahI(n,y) = {sum(k=1, n, y^k*(-1)^(n-k)*(n!/k!)*binomial(n-1, k-1))}
S(n) = {my(p=prod(k=1, n, 1 + y*x^k + O(x*x^n))); 1 + sum(i=1, (sqrtint(8*n+1)-1)\2, polcoef(p,i,y)*LahI(i,y))}
seq(n)={my(q=S(n)); [subst(serlaplace(p),y,1) | p<-Vec(prod(k=1, n, subst(q + O(x*x^(n\k)), x, x^k)))]} \\ Andrew Howroyd, Feb 12 2022

Terms a(26) and beyond from Andrew Howroyd, Feb 12 2022

A328595 Numbers whose reversed binary expansion is a necklace.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 12, 14, 15, 16, 20, 24, 26, 28, 30, 31, 32, 36, 40, 42, 44, 48, 52, 54, 56, 58, 60, 62, 63, 64, 72, 80, 84, 88, 92, 96, 100, 104, 106, 108, 112, 116, 118, 120, 122, 124, 126, 127, 128, 136, 144, 152, 160, 164, 168, 170, 172, 176, 180

Offset: 1

Gus Wiseman, Oct 22 2019

A necklace is a finite sequence that is lexicographically minimal among all of its cyclic rotations.

			The sequence of terms together with their binary expansions and binary indices begins:
   1:      1 ~ {1}
   2:     10 ~ {2}
   3:     11 ~ {1,2}
   4:    100 ~ {3}
   6:    110 ~ {2,3}
   7:    111 ~ {1,2,3}
   8:   1000 ~ {4}
  10:   1010 ~ {2,4}
  12:   1100 ~ {3,4}
  14:   1110 ~ {2,3,4}
  15:   1111 ~ {1,2,3,4}
  16:  10000 ~ {5}
  20:  10100 ~ {3,5}
  24:  11000 ~ {4,5}
  26:  11010 ~ {2,4,5}
  28:  11100 ~ {3,4,5}
  30:  11110 ~ {2,3,4,5}
  31:  11111 ~ {1,2,3,4,5}
  32: 100000 ~ {6}
  36: 100100 ~ {3,6}

John Tyler Rascoe, Table of n, a(n) for n = 1..10000

A similar concept is A065609.
The version with the most significant digit ignored is A328607.
Lyndon words are A328596.
Aperiodic words are A328594.
Binary necklaces are A000031.
Necklace compositions are A008965.
Cf. A000120, A000740, A001037, A032153, A059966, A275692, A328668.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Select[Range[100],neckQ[Reverse[IntegerDigits[#,2]]]&]

from itertools import count, islice
from sympy.utilities.iterables import necklaces
def a_gen():
    for n in count(1):
        t = []
        for i in necklaces(n,2):
            if sum(i)>0:
                t.append(sum(2**j for j in range(len(i)) if i[j] > 0))
        yield from sorted(t)
A328595_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, May 24 2024

A065608 Sum of divisors of n minus the number of divisors of n.

Original entry on oeis.org

0, 1, 2, 4, 4, 8, 6, 11, 10, 14, 10, 22, 12, 20, 20, 26, 16, 33, 18, 36, 28, 32, 22, 52, 28, 38, 36, 50, 28, 64, 30, 57, 44, 50, 44, 82, 36, 56, 52, 82, 40, 88, 42, 78, 72, 68, 46, 114, 54, 87, 68, 92, 52, 112, 68, 112, 76, 86, 58, 156, 60, 92, 98, 120, 80, 136, 66, 120, 92

Offset: 1

Jason Earls, Nov 06 2001

Number of permutations p of {1,2,...,n} such that p(k)-k takes exactly two distinct values. Example: a(4)=4 because we have 4123, 3412, 2143 and 2341. - Max Alekseyev and Emeric Deutsch, Dec 22 2006
Number of solutions to the Diophantine equation xy + yz = n, with x,y,z >= 1.
In other words, number of ways to write n = (a + b) * k for positive integers a, b, k. - Gus Wiseman, Mar 25 2021
Not the same as A184396(n): a(66) = 136 while A184396(66) = 137. - Wesley Ivan Hurt, Dec 26 2013
From Gus Wiseman, Mar 25 2021: (Start)
Also the number of compositions of n into an even number of parts with alternating parts equal. These are finite even-length sequences q of positive integers summing to n such that q(i) = q(i+2) for all possible i. For example, the a(2) = 1 through a(8) = 11 compositions are:
  (1,1)  (1,2)  (1,3)      (1,4)  (1,5)          (1,6)  (1,7)
         (2,1)  (2,2)      (2,3)  (2,4)          (2,5)  (2,6)
                (3,1)      (3,2)  (3,3)          (3,4)  (3,5)
                (1,1,1,1)  (4,1)  (4,2)          (4,3)  (4,4)
                                  (5,1)          (5,2)  (5,3)
                                  (1,2,1,2)      (6,1)  (6,2)
                                  (2,1,2,1)             (7,1)
                                  (1,1,1,1,1,1)         (1,3,1,3)
                                                        (2,2,2,2)
                                                        (3,1,3,1)
                                                        (1,1,1,1,1,1,1,1)
The odd-length version is A062968.
The version with alternating parts weakly decreasing is A114921, or A342528 if odd-length compositions are included.
The version with alternating parts unequal is A342532, or A224958 if odd-length compositions are included (unordered: A339404/A000726).
Allowing odd lengths as well as even gives A342527.
(End)
Inverse Möbius transform of n-1. - Wesley Ivan Hurt, Jun 29 2024

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
M. Alekseyev, E. Deutsch, and J. H. Steelman, Solution to problem 11281, Amer. Math. Monthly, 116, No. 5, 2009, p. 465.
George E. Andrews, Stacked lattice boxes, Ann. Comb. 3 (1999), 115-130. See L_2(n).
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
Masato Kobayashi, New recurrences for divisor sum functions and triangular numbers, arXiv:2207.05831 [math.NT], 2022.

Cf. A000203, A000005, A134857.
Starting (1, 2, 4, 4, 8, 6, ...), = row sums of triangle A077478. - Gary W. Adamson, Nov 12 2007
Starting with "1" = row sums of triangle A176919. - Gary W. Adamson, Apr 29 2010
Column k=2 of A125182.
A175342/A325545 count compositions with constant/distinct differences.
Cf. A001522, A002843, A008965, A010054, A064410, A064428, A325557, A342495.

List([1..100],n->Sigma(n)-Tau(n)); # Muniru A Asiru, Mar 19 2018

with(numtheory): seq(sigma(n)-tau(n),n=1..70); # Emeric Deutsch, Dec 22 2006

Table[DivisorSigma[1,n]-DivisorSigma[0,n], {n,100}] (* Wesley Ivan Hurt, Dec 26 2013 *)

a(n) = sigma(n) - numdiv(n); \\ Harry J. Smith, Oct 23 2009

from math import prod
from sympy import factorint
def A065608(n):
    f = factorint(n).items()
    return prod((p**(e+1)-1)//(p-1) for p, e in f)-prod(e+1 for p,e in f) # Chai Wah Wu, Jul 16 2022

a(n) = sigma(n) - d(n) = A000203(n) - A000005(n).
a(n) = Sum_{d|n} (d-1). - Wesley Ivan Hurt, Dec 26 2013
G.f.: Sum_{k>=1} x^(2*k)/(1-x^k)^2. - Benoit Cloitre, Apr 21 2003
G.f.: Sum_{n>=1} (n-1)*x^n/(1-x^n). - Joerg Arndt, Jan 30 2011
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(1-1/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 18 2018
G.f.: Sum_{n >= 1} q^(n^2)*( (n - 1) + q^n - (n - 1)*q^(2*n) )/(1 - q^n)^2 - differentiate equation 1 in Arndt with respect to t, then set x = q and t = q. - Peter Bala, Jan 22 2021
a(n) = A342527(n) - A062968(n). - Gus Wiseman, Mar 25 2021
a(n) = n * A010054(n) - Sum_{k>=1} a(n - k*(k+1)/2), assuming a(n) = 0 for n <= 0 (Kobayashi, 2022). - Amiram Eldar, Jun 23 2023

A065609 Positive m such that when written in binary, no rotated value of m is greater than m.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 12, 14, 15, 16, 20, 24, 26, 28, 30, 31, 32, 36, 40, 42, 48, 50, 52, 54, 56, 58, 60, 62, 63, 64, 72, 80, 84, 96, 98, 100, 104, 106, 108, 112, 114, 116, 118, 120, 122, 124, 126, 127, 128, 136, 144, 160, 164, 168, 170, 192, 194, 196, 200, 202

Offset: 1

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Nov 06 2001

Rotated values of m are defined as the numbers which occur when m is shifted 1, 2, ... bits to the right with the last bits added to the front; e.g., the rotated values of 1011 are 1011, 1101, 1110 and 0111.
The number of k-bit binary numbers in this sequence is A008965. This gives the row lengths when the sequence is regarded as a table.
If m is in the sequence, then so is 2m. All odd terms are of the form 2^k - 1. - Ivan Neretin, Aug 04 2016
First differs from A328595 in lacking 44, with binary expansion {1, 0, 1, 1, 0, 0}, and 92, with binary expansion {1, 0, 1, 1, 1, 0, 0}. - Gus Wiseman, Oct 31 2019

			14 is included because 14 in binary is 1110. 1110 has the rotated values of 0111, 1011 and 1101 -- 7, 11 and 13 -- which are all smaller than 14.

Ivan Neretin, Table of n, a(n) for n = 1..10000

A similar concept is A328595.
The version with the most significant digit ignored is A328668 or A328607.
Numbers whose reversed binary expansion is a Lyndon word are A328596.
Numbers whose binary expansion is aperiodic are A328594.
Binary necklaces are A000031.
Necklace compositions are A008965.
Cf. A000031, A000120, A000740, A001037, A032153, A059966, A163381, A275692.

filter:= proc(n) local L, k;
  if n::odd then return evalb(n+1 = 2^ilog2(n+1)) fi;
  L:= convert(convert(n,binary),string);
  for k from 1 to length(L)-1 do
    if not lexorder(StringTools:-Rotate(L,k),L) then return false fi;
  od;
  true
end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 05 2016

Select[Range[200], # == Max[FromDigits[#, 2] & /@ NestList[RotateLeft, dg = IntegerDigits[#, 2], Length@dg]] &] (* Ivan Neretin, Aug 04 2016 *)

def ok(n):
    b = bin(n)[2:]
    return b > "0" and all(b[i:] + b[:i] <= b for i in range(1, len(b)))
print([k for k in range(203) if ok(k)]) # Michael S. Branicky, May 26 2022

Edited by Franklin T. Adams-Watters, Apr 09 2010

A344615 Number of compositions of n with no adjacent triples (..., x, y, z, ...) where x <= y <= z.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 17, 29, 50, 84, 143, 241, 408, 688, 1162, 1959, 3305, 5571, 9393, 15832, 26688, 44980, 75812, 127769, 215338, 362911, 611620, 1030758, 1737131, 2927556, 4933760, 8314754, 14012668, 23615198, 39798098, 67070686, 113032453, 190490542, 321028554

Offset: 0

Gus Wiseman, May 27 2021

nonn

These compositions avoid the weak consecutive pattern (1,2,3), the strict version being A128761.

			The a(1) = 1 through a(6) = 17 compositions:
  (1)  (2)    (3)    (4)      (5)        (6)
       (1,1)  (1,2)  (1,3)    (1,4)      (1,5)
              (2,1)  (2,2)    (2,3)      (2,4)
                     (3,1)    (3,2)      (3,3)
                     (1,2,1)  (4,1)      (4,2)
                     (2,1,1)  (1,3,1)    (5,1)
                              (2,1,2)    (1,3,2)
                              (2,2,1)    (1,4,1)
                              (3,1,1)    (2,1,3)
                              (1,2,1,1)  (2,3,1)
                                         (3,1,2)
                                         (3,2,1)
                                         (4,1,1)
                                         (1,2,1,2)
                                         (1,3,1,1)
                                         (2,1,2,1)
                                         (2,2,1,1)

The case of permutations is A049774.
The strict non-adjacent version is A102726.
The case of permutations of prime indices is A344652.
A001250 counts alternating permutations.
A005649 counts anti-run patterns.
A106356 counts compositions by number of maximal anti-runs.
A114901 counts compositions where each part is adjacent to an equal part.
A344604 counts wiggly compositions with twins.
A344605 counts wiggly patterns with twins.
A344606 counts wiggly permutations of prime factors with twins.
Counting compositions by patterns:
- A003242 avoiding (1,1) adjacent.
- A011782 no conditions.
- A106351 avoiding (1,1) adjacent by sum and length.
- A128695 avoiding (1,1,1) adjacent.
- A128761 avoiding (1,2,3).
- A232432 avoiding (1,1,1).
- A335456 all patterns.
- A335457 all patterns adjacent.
- A335514 matching (1,2,3).
- A344604 weakly avoiding (1,2,3) and (3,2,1) adjacent.
- A344614 avoiding (1,2,3) and (3,2,1) adjacent.
- A344615 weakly avoiding (1,2,3) adjacent.
Cf. A000041, A006330, A008965, A027187, A238279/A333755, A333213, A335464, A335515, A344612, A344619.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z]&]],{n,0,15}]

More terms from Bert Dobbelaere, Jun 12 2021

A069916 Number of log-concave compositions (ordered partitions) of n.

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 14, 20, 26, 36, 47, 60, 80, 102, 127, 159, 194, 236, 291, 355, 425, 514, 611, 718, 856, 1009, 1182, 1381, 1605, 1861, 2156, 2496, 2873, 3299, 3778, 4301, 4902, 5574, 6325, 7176, 8116, 9152, 10317, 11610, 13028, 14611, 16354, 18259, 20365

Offset: 0

Pontus von Brömssen, Apr 24 2002

These are compositions with weakly decreasing first quotients, where the first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3). - Gus Wiseman, Mar 16 2021

			Out of the 8 compositions of 4, only 2+1+1 and 1+1+2 are not log-concave, so a(4)=6.
From _Gus Wiseman_, Mar 15 2021: (Start)
The a(1) = 1 through a(6) = 14 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (13)    (14)     (15)
             (21)   (22)    (23)     (24)
             (111)  (31)    (32)     (33)
                    (121)   (41)     (42)
                    (1111)  (122)    (51)
                            (131)    (123)
                            (221)    (132)
                            (11111)  (141)
                                     (222)
                                     (231)
                                     (321)
                                     (1221)
                                     (111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..300
Sean A. Irvine, Java program (github)
Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version for differences instead of quotients is A070211.
A000005 counts constant compositions.
A000009 counts strictly increasing (or strictly decreasing) compositions.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A001055 counts factorizations.
A002843 counts compositions with adjacent parts x <= 2y.
A003238 counts chains of divisors summing to n-1, with strict case A122651.
A003242 counts anti-run compositions.
A074206 counts ordered factorizations.
A167865 counts strict chains of divisors summing to n.
Cf. A008965, A048004, A059966, A167606, A175342, A325547.

(* This program is not suitable for computing a large number of terms *)
compos[n_] := Permutations /@ IntegerPartitions[n] // Flatten[#, 1]&;
logConcaveQ[p_] := And @@ Table[p[[i]]^2 >= p[[i-1]]*p[[i+1]], {i, 2, Length[p]-1}]; a[n_] := Count[compos[n], p_?logConcaveQ]; Table[an = a[n]; Print["a(", n, ") = ", an]; a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 29 2016 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],GreaterEqual@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,15}] (* Gus Wiseman, Mar 15 2021 *)

def A069916(n) : return sum(all(p[i]^2 >= p[i-1] * p[i+1] for i in range(1, len(p)-1)) for p in Compositions(n)) # Eric M. Schmidt, Sep 29 2013

A184271 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal binary arrays (n >= 1, k >= 1).

Original entry on oeis.org

2, 3, 3, 4, 7, 4, 6, 14, 14, 6, 8, 40, 64, 40, 8, 14, 108, 352, 352, 108, 14, 20, 362, 2192, 4156, 2192, 362, 20, 36, 1182, 14624, 52488, 52488, 14624, 1182, 36, 60, 4150, 99880, 699600, 1342208, 699600, 99880, 4150, 60, 108, 14602, 699252, 9587580, 35792568

Offset: 1

R. H. Hardin, Jan 10 2011

This is a 2-dimensional generalization of binary necklaces (A000031). A toroidal array or necklace can be defined either as an equivalence class of matrices under all possible rotations of the sequence of rows and the sequence of columns, or as a matrix that is minimal among all possible rotations of its sequence of rows and its sequence of columns. - Gus Wiseman, Feb 04 2019

			      1     2        3           4            5             6              7
----------------------------------------------------------------------------
1:    2     3        4           6            8            14             20
2:    3     7       14          40          108           362           1182
3:    4    14       64         352         2192         14624          99880
4:    6    40      352        4156        52488        699600        9587580
5:    8   108     2192       52488      1342208      35792568      981706832
6:   14   362    14624      699600     35792568    1908897152   104715443852
7:   20  1182    99880     9587580    981706832  104715443852 11488774559744
8:   36  4150   699252   134223976  27487816992 5864063066500
9:   60 14602  4971184  1908881900 781874936816
10: 108 52588 35792568 27487869472
From _Gus Wiseman_, Feb 04 2019: (Start)
Inequivalent representatives of the T(2,3) = 14 toroidal necklaces:
  [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 1] [0 0 1] [0 0 1]
  [0 0 0] [0 0 1] [0 1 1] [1 1 1] [0 0 1] [0 1 0] [0 1 1]
.
  [0 0 1] [0 0 1] [0 0 1] [0 1 1] [0 1 1] [0 1 1] [1 1 1]
  [1 0 1] [1 1 0] [1 1 1] [0 1 1] [1 0 1] [1 1 1] [1 1 1]
(End)

Alois P. Heinz, Antidiagonals n = 1..100, flattened (first 95 terms from R. H. Hardin)
S. N. Ethier, Counting toroidal binary arrays, arXiv preprint arXiv:1301.2352, 2013 and J. Int. Seq. 16 (2013) #13.4.7 .
S. N. Ethier and Jiyeon Lee, Counting toroidal binary arrays, II, arXiv:1502.03792v1 [math.CO], Feb 12, 2015 and J. Int. Seq. 18 (2015) # 15.8.3.
Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023.
Wikipedia, Pólya enumeration theorem

Columns 1-8 are A000031, A184264, A184265, A184266, A184267, A184268, A184269, A184270.
Main diagonal is A179043.
Cf. A001037 (binary Lyndon words), A008965, A323858, A323859 (binary toroidal necklaces of size n), A323861 (aperiodic version), A323865, A323870 (normal toroidal necklaces), A323872.

a[n_, k_] := Sum[If[Mod[n, c] == 0, Sum[If[Mod[k, d] == 0, EulerPhi[c] EulerPhi[d] 2^(n k/LCM[c, d]), 0], {d, 1, k}], 0], {c, 1, n}]/(n k)
(* second program *)
neckmatQ[m_]:=m==First[Union@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}]];
Table[Length[Select[Partition[#,n-k]&/@Tuples[{0,1},(n-k)*k],neckmatQ]],{n,8},{k,n-1}] (* Gus Wiseman, Feb 04 2019 *)

T(n,k) = (1/(nk))*Sum_{ c divides n } Sum_{ d divides k } phi(c)*phi(d)*2^(nk/lcm(c,d)), where phi is A000010 and lcm stands for least common multiple. - Stewart N. Ethier, Aug 24 2012

A329398 Number of compositions of n with uniform Lyndon factorization and uniform co-Lyndon factorization.

Original entry on oeis.org

1, 2, 4, 7, 12, 18, 28, 40, 57, 80, 110, 148, 200, 266, 348, 457, 592, 764, 978, 1248, 1580, 2000, 2508, 3142, 3913

Offset: 1

Gus Wiseman, Nov 13 2019

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 word is a finite sequence that is prime with respect to the Lyndon product. Equivalently, a Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into Lyndon words, and if these factors are arranged in lexicographically decreasing order, their concatenation is equal to their Lyndon product. For example, (1001) has sorted Lyndon factorization (001)(1).
Similarly, the co-Lyndon product is the lexicographically minimal sequence obtainable by shuffling the sequences together, and a co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product, or, equivalently, a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. For example, (1001) has sorted co-Lyndon factorization (1)(100).
A sequence of words is uniform if they all have the same length.
Conjecture: Also the number of compositions of n that are either weakly increasing or weakly decreasing. Hence a(n) = 2 * A000041(n) - A000005(n). - Gus Wiseman, Mar 05 2020

			The a(1) = 1 through a(6) = 18 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (13)    (14)     (15)
             (21)   (22)    (23)     (24)
             (111)  (31)    (32)     (33)
                    (112)   (41)     (42)
                    (211)   (113)    (51)
                    (1111)  (122)    (114)
                            (221)    (123)
                            (311)    (222)
                            (1112)   (321)
                            (2111)   (411)
                            (11111)  (1113)
                                     (1122)
                                     (2211)
                                     (3111)
                                     (11112)
                                     (21111)
                                     (111111)

Lyndon and co-Lyndon compositions are (both) counted by A059966.
Lyndon compositions that are not weakly increasing are A329141.
Lyndon compositions whose reverse is not co-Lyndon are A329324.
Cf. A000740, A001037, A001523, A008965, A059204, A060223, A211100, A328596, A329312, A329318, A329396, A329397, A329399, A332578, A332669, A332834.

lynQ[q_]:=Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
lynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[lynfac[Drop[q,i]],Take[q,i]]][Last[Select[Range[Length[q]],lynQ[Take[q,#]]&]]]];
colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
colynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[colynfac[Drop[q,i]],Take[q,i]]]@Last[Select[Range[Length[q]],colynQ[Take[q,#]]&]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@Length/@lynfac[#]&&SameQ@@Length/@colynfac[#]&]],{n,10}]

a(19)-a(25) from Robert Price, Jun 20 2021

A349052 Number of weakly alternating compositions of n.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 28, 52, 91, 161, 280, 491, 850, 1483, 2573, 4469, 7757, 13472, 23378, 40586, 70438, 122267, 212210, 368336, 639296, 1109620, 1925916, 3342755, 5801880, 10070133, 17478330, 30336518, 52653939, 91389518, 158621355, 275313226, 477850887, 829388075

Offset: 0

Gus Wiseman, Nov 29 2021

nonn

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. A sequence is alternating iff it is a weakly alternating anti-run.

			The a(5) = 16 compositions:
  (1,1,1,1,1)  (1,1,1,2)  (1,1,3)  (1,4)  (5)
               (1,1,2,1)  (1,2,2)  (2,3)
               (1,2,1,1)  (1,3,1)  (3,2)
               (2,1,1,1)  (2,1,2)  (4,1)
                          (2,2,1)
                          (3,1,1)
The a(6) = 28 compositions:
  (111111)  (11112)  (1113)  (114)  (15)  (6)
            (11121)  (1122)  (132)  (24)
            (11211)  (1131)  (141)  (33)
            (12111)  (1212)  (213)  (42)
            (21111)  (1311)  (222)  (51)
                     (2121)  (231)
                     (2211)  (312)
                     (3111)  (411)

Andrew Howroyd, Table of n, a(n) for n = 0..1000 (Terms 0..55 from Martin Ehrenstein)

The strong case is A025047, ranked by A345167.
The directed versions are A129852 and A129853, strong A025048 and A025049.
The complement is counted by A349053, strong A345192.
The version for permutations of prime indices is A349056, strong A345164.
The complement is ranked by A349057, strong A345168.
The version for patterns is A349058, strong A345194.
The multiplicative version is A349059, strong A348610.
An unordered version (partitions) is A349060, complement A349061.
The non-alternating case is A349800, ranked by A349799.
A001250 counts alternating permutations, complement A348615.
A001700 counts compositions of 2n with alternating sum 0.
A003242 counts Carlitz (anti-run) compositions.
A011782 counts compositions.
A106356 counts compositions by number of maximal anti-runs.
A344604 counts alternating compositions with twins.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A349054 counts strict alternating compositions.
Cf. A000041, A008965, A102726, A114901, A128761, A261983, A333213, A333755, A344614, A344615, A345165, A345195.

whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],whkQ[#]||whkQ[-#]&]],{n,0,10}]

C(n,f)={my(M=matrix(n,n,j,k,k>=j), s=M[,n]); for(b=1, n, f=!f; M=matrix(n,n,j,k, if(k1,M[j-k,k-1]) ))); for(k=2, n, M[,k]+=M[,k-1]); s+=M[,n]); s~}
seq(n) = concat([1], C(n,0) + C(n,1) - vector(n,j,numdiv(j))) \\ Andrew Howroyd, Jan 31 2024

a(21)-a(37) from Martin Ehrenstein, Jan 08 2022

cp's OEIS Frontend

A345168 Numbers k such that the k-th composition in standard order is not alternating.

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A351013 Number of integer compositions of n with all distinct runs.

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A328595 Numbers whose reversed binary expansion is a necklace.

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A065608 Sum of divisors of n minus the number of divisors of n.

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A065609 Positive m such that when written in binary, no rotated value of m is greater than m.

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A344615 Number of compositions of n with no adjacent triples (..., x, y, z, ...) where x <= y <= z.

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A069916 Number of log-concave compositions (ordered partitions) of n.

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