A008965 -id:A008965 - cp's OEIS frontend

A329745 Number of compositions of n with runs-resistance 2.

0, 0, 2, 3, 6, 15, 22, 41, 72, 129, 213, 395, 660, 1173, 2031, 3582, 6188, 10927, 18977, 33333, 58153, 101954, 178044, 312080, 545475, 955317, 1670990, 2925711, 5118558, 8960938, 15680072, 27447344, 48033498, 84076139, 147142492, 257546234, 450748482, 788937188

Offset: 1

Gus Wiseman, Nov 21 2019

nonn

A composition of n is a finite sequence of positive integers with sum n.
For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.
These are non-constant compositions with equal run-lengths (A329738).

			The a(3) = 2 through a(6) = 15 compositions:
  (1,2)  (1,3)    (1,4)    (1,5)
  (2,1)  (3,1)    (2,3)    (2,4)
         (1,2,1)  (3,2)    (4,2)
                  (4,1)    (5,1)
                  (1,3,1)  (1,2,3)
                  (2,1,2)  (1,3,2)
                           (1,4,1)
                           (2,1,3)
                           (2,3,1)
                           (3,1,2)
                           (3,2,1)
                           (1,1,2,2)
                           (1,2,1,2)
                           (2,1,2,1)
                           (2,2,1,1)

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Column k = 2 of A329744.
Column k = n - 2 of A329750.
Cf. A000740, A003242, A008965, A098504, A242882, A318928, A329743, A329746, A329747, A329767.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]==2&]],{n,10}]

seq(n)={my(b=Vec(1/(1 - sum(k=1, n, x^k/(1+x^k) + O(x*x^n)))-1)); vector(n, k, sumdiv(k, d, b[d]-1))} \\ Andrew Howroyd, Dec 30 2020

a(n) = A329738(n) - A000005(n).

a(n) = Sum_{d|n} (A003242(d) - 1). - Andrew Howroyd, Dec 30 2020

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020

A342192 Heinz numbers of partitions of crank 0.

Original entry on oeis.org

6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 100, 106, 118, 122, 134, 140, 142, 146, 158, 166, 178, 194, 196, 202, 206, 214, 218, 220, 226, 254, 260, 262, 274, 278, 298, 300, 302, 308, 314, 326, 334, 340, 346, 358, 362, 364, 380, 382, 386, 394, 398

Offset: 1

Gus Wiseman, Apr 05 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

See A257989 or the program for a definition of crank of a partition.

			The sequence of terms together with their prime indices begins:
      6: {1,2}        106: {1,16}       218: {1,29}
     10: {1,3}        118: {1,17}       220: {1,1,3,5}
     14: {1,4}        122: {1,18}       226: {1,30}
     22: {1,5}        134: {1,19}       254: {1,31}
     26: {1,6}        140: {1,1,3,4}    260: {1,1,3,6}
     34: {1,7}        142: {1,20}       262: {1,32}
     38: {1,8}        146: {1,21}       274: {1,33}
     46: {1,9}        158: {1,22}       278: {1,34}
     58: {1,10}       166: {1,23}       298: {1,35}
     62: {1,11}       178: {1,24}       300: {1,1,2,3,3}
     74: {1,12}       194: {1,25}       302: {1,36}
     82: {1,13}       196: {1,1,4,4}    308: {1,1,4,5}
     86: {1,14}       202: {1,26}       314: {1,37}
     94: {1,15}       206: {1,27}       326: {1,38}
    100: {1,1,3,3}    214: {1,28}       334: {1,39}

Indices of zeros in A257989.
A000005 counts constant partitions.
A000041 counts partitions (strict: A000009).
A001522 counts partitions of positive crank.
A003242 counts anti-run compositions.
A064391 counts partitions by crank.
A064428 counts partitions of nonnegative crank.
A224958 counts compositions with alternating parts unequal.
A257989 gives the crank of the partition with Heinz number n.
Cf. A000726, A008965, A056239, A112798, A124010, A130091, A325351, A325352.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ck[y_]:=With[{w=Count[y,1]},If[w==0,Max@@y,Count[y,_?(#>w&)]-w]];
Select[Range[100],ck[primeMS[#]]==0&]

A329766 Number of compositions of n whose run-lengths cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 1, 3, 6, 13, 21, 48, 89, 180, 355, 707, 1382, 2758, 5448, 10786, 21391, 42476, 84291, 167516, 333036, 662153, 1317687, 2622706, 5221951, 10400350, 20720877, 41288823, 82294979, 164052035, 327088649, 652238016, 1300788712, 2594486045, 5175378128, 10324522020

Offset: 0

Gus Wiseman, Nov 20 2019

nonn

A composition of n is a finite sequence of positive integers with sum n.

			The a(0) = 1 through a(5) = 13 compositions:
  ()  (1)  (2)  (3)    (4)      (5)
                (1,2)  (1,3)    (1,4)
                (2,1)  (3,1)    (2,3)
                       (1,1,2)  (3,2)
                       (1,2,1)  (4,1)
                       (2,1,1)  (1,1,3)
                                (1,2,2)
                                (1,3,1)
                                (2,1,2)
                                (2,2,1)
                                (3,1,1)
                                (1,1,2,1)
                                (1,2,1,1)

Looking at multiplicities instead of run-lengths gives A329741.
The complete case is A329749.
Complete compositions are A107429.
Cf. A000740, A008965, A242882, A244164, A274174, A329738, A329739, A329740, A329744, A329748.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],normQ[Length/@Split[#]]&]],{n,0,10}]

a(21)-a(26) from Giovanni Resta, Nov 22 2019

a(27)-a(35) from Alois P. Heinz, Jul 06 2020

A333943 Numbers k such that the k-th composition in standard order is a reversed necklace.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 15, 16, 17, 18, 19, 21, 23, 31, 32, 33, 34, 35, 36, 37, 39, 41, 42, 43, 45, 47, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 77, 79, 81, 83, 85, 87, 91, 95, 127, 128, 129, 130, 131, 132, 133, 135, 136, 137, 138, 139, 141, 143

Offset: 1

Gus Wiseman, Apr 14 2020

nonn

A necklace is a finite sequence that is lexicographically minimal among all of its cyclic rotations. Reversed necklaces are different from co-necklaces (A333764).

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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 sequence together with the corresponding reversed necklaces begins:
    1: (1)             32: (6)               69: (4,2,1)
    2: (2)             33: (5,1)             71: (4,1,1,1)
    3: (1,1)           34: (4,2)             73: (3,3,1)
    4: (3)             35: (4,1,1)           74: (3,2,2)
    5: (2,1)           36: (3,3)             75: (3,2,1,1)
    7: (1,1,1)         37: (3,2,1)           77: (3,1,2,1)
    8: (4)             39: (3,1,1,1)         79: (3,1,1,1,1)
    9: (3,1)           41: (2,3,1)           81: (2,4,1)
   10: (2,2)           42: (2,2,2)           83: (2,3,1,1)
   11: (2,1,1)         43: (2,2,1,1)         85: (2,2,2,1)
   15: (1,1,1,1)       45: (2,1,2,1)         87: (2,2,1,1,1)
   16: (5)             47: (2,1,1,1,1)       91: (2,1,2,1,1)
   17: (4,1)           63: (1,1,1,1,1,1)     95: (2,1,1,1,1,1)
   18: (3,2)           64: (7)              127: (1,1,1,1,1,1,1)
   19: (3,1,1)         65: (6,1)            128: (8)
   21: (2,2,1)         66: (5,2)            129: (7,1)
   23: (2,1,1,1)       67: (5,1,1)          130: (6,2)
   31: (1,1,1,1,1)     68: (4,3)            131: (6,1,1)

The non-reversed version is A065609.
The dual version is A328595.
Binary necklaces are A000031.
Necklace compositions are A008965.
Necklaces covering an initial interval are A019536.
Numbers whose prime signature is a necklace are A329138.
Length of co-Lyndon factorization of binary expansion is A329312.
Length of Lyndon factorization of reversed binary expansion is A329313.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Runs are counted by A124767.
- Rotational symmetries are counted by A138904.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon compositions are A275692.
- Co-Lyndon compositions are A326774.
- Aperiodic compositions are A328594.
- Length of Lyndon factorization is A329312.
- Rotational period is A333632.
- Co-necklaces are A333764.
- Length of co-Lyndon factorization is A334029.
Cf. A000740, A001037, A027375, A059966, A211100, A302291, A328596, A329142, A333765, A333939, A333941.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#1]}]&,Length[q]-1,1,And];
Select[Range[100],neckQ[Reverse[stc[#]]]&]

A037306 Triangle T(n,k) read by rows: the number of compositions of n into k parts, modulo cyclic shifts.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 4, 3, 1, 1, 1, 3, 5, 5, 3, 1, 1, 1, 4, 7, 10, 7, 4, 1, 1, 1, 4, 10, 14, 14, 10, 4, 1, 1, 1, 5, 12, 22, 26, 22, 12, 5, 1, 1, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1, 1, 6, 19, 43, 66, 80, 66, 43, 19, 6, 1, 1, 1, 6, 22, 55, 99, 132, 132, 99, 55, 22, 6, 1

Offset: 1

Jens Voß, Jun 30 2001

Triangle obtained from A047996 by dropping the first column (k=0), and row (n=0).
T(n, k) = number of different ways the number n can be expressed as ordered sums of k positive integers, counting only once those ordered sums that can be transformed into each other by a cyclic permutation.
These might be described as cyclic compositions, or more loosely as cyclic partitions. - N. J. A. Sloane, Sep 05 2012

			Triangle begins
  1;
  1,  1;
  1,  1,  1;
  1,  2,  1,  1;
  1,  2,  2,  1,  1;
  1,  3,  4,  3,  1,  1;
  1,  3,  5,  5,  3,  1,  1;
  1,  4,  7, 10,  7,  4,  1,  1;
  1,  4, 10, 14, 14, 10,  4,  1,  1;
  1,  5, 12, 22, 26, 22, 12,  5,  1,  1;
  1,  5, 15, 30, 42, 42, 30, 15,  5,  1,  1;
T(6,3) = 4, since there are 4 essentially different ways 1+1+4, 1+2+3, 1+3+2 and 2+2+2 of expressing 6 as a sum of 3 summands (all others can be obtained by cyclically permuting the summands in one of the above sums).

N. Zagaglia Salvi, Ordered partitions and colourings of cycles and necklaces, Bull. Inst. Combin. Appl., 27 (1999), 37-40.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
Ethan Akin and Morton Davis, Bulgarian solitaire, Am. Math. Monthly 92 (4) (1985) 237-250.
R. Baumann, Computer-Knobelei, , LOGIN, 163/164 (2010), 141-142 (in German).
R. Bekes, J. Pedersen and B. Shao, Mad tea party cyclic partitions, College Math. J., 43 (2012), 24-36.
A. Elashvili and M. Jibladze, Hermite reciprocity for the regular representations of cyclic groups, Indag. Math. (N.S.) 9 (1998), no. 2, 233-238. MR1691428 (2000c:13006)
A. Elashvili, M. Jibladze, and D. Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Algebraic Combin. 10 (1999), no. 2, 173-188. MR1719140 (2000j:05009). See p. 174. - _N. J. A. Sloane_, Aug 06 2014
Petros Hadjicostas, Cyclic Compositions of a Positive Integer with Parts Avoiding an Arithmetic Sequence, Journal of Integer Sequences, 19 (2016), Article 16.8.2.
Arnold Knopfmacher and Neville Robbins, Some Properties of Cyclic Compositions, Fibonacci Quart. 48 (2010), no. 3, 249-255.
D. M. Y. Sommerville, On Certain Periodic Properties of Cyclic Compositions of Numbers, Proc. London Math. Soc., s2-7, No. 1 (1909), 263-313.
R. Razen, J. Seberry, and K. Wehrhahn, Ordered partitions and codes generated by circulant matrices, J. Combin. Theory Ser. A, 27 (1979), 333-341.
D. Wasserman, Proof of the symmetry [Broken link]
D. Wasserman, Proof of the symmetry [Cached copy]

A047996 and A241926 are essentially identical to this entry.
Cf. A008965 (row sums), A000010, A007318, A027750, A215251, A004526 (column 2), A007997 (column 3), A008610 (column 4), A008646 (column 5), A032191 (column 6).
See A245558, A245559 for a closely related array.
See A052307 for compositions modulo cyclic shifts and reversal.

a037306 n k = div (sum $ map f $ a027750_row $ gcd n k) n where
   f d = a000010 d * a007318' (div n d) (div k d)
a037306_row n = map (a037306 n) [1..n]
a037306_tabl = map a037306_row [1..]
-- Reinhard Zumkeller, Feb 06 2014

A037306 := proc(n,k) local a,d; a := 0; for d in numtheory[divisors]( igcd(n,k)) do a := a+numtheory[phi](d)*binomial(n/d,k/d); end do: a/n; end proc:
seq(seq(A037306(n,k), k=1..n), n=1..20); # R. J. Mathar, Jun 11 2011

t[n_, k_] := Total[EulerPhi[#]*Binomial[n/#, k/#] & /@ Divisors[GCD[n, k]]]/n; Flatten[Table[t[n, k], {n, 13}, {k, n}]] (* Jean-François Alcover, Sep 08 2011, after formula *)
nn=15;f[list_]:=Select[list,#>0&];Map[f,Transpose[Table[Drop[CoefficientList[Series[CycleIndex[CyclicGroup[n],s]/.Table[s[i]->x^i/(1-x^i),{i,1,n}],{x,0,nn}],x],1],{n,1,nn}]]]//Grid  (* Geoffrey Critzer, Oct 30 2012 *)

T(n, k) = sumdiv(gcd(n,k), d, eulerphi(d)*binomial(n/d, k/d))/n; \\ Michel Marcus, Feb 10 2016

T(n,k) = (Sum_{d|gcd(n,k)} phi(d)*binomial(n/d,k/d))/n, where phi = A000010 = Euler's totient function. Also T(n,k) = A047996(n,k). - Paul Weisenhorn, Apr 06 2011

More terms from David Wasserman, Mar 11 2002

Comments, reference, example from Paul Weisenhorn, Dec 18 2010

A179043 Number of n X n checkered tori.

Original entry on oeis.org

1, 2, 7, 64, 4156, 1342208, 1908897152, 11488774559744, 288230376353050816, 29850020237398264483840, 12676506002282327791964489728, 21970710674130840874443091905462272, 154866286100907105149651981766316633972736

Offset: 0

Rouben Rostamian (rostamian(AT)umbc.edu), Jun 25 2010

nonn

Consider an n X n checkerboard whose tiles are assigned colors 0 and 1, at random. There are 2^(n^2) such checkerboards. We identify the opposite edges of each checkerboard, thus making it into a (topological) torus. There are a(n) such (distinct) tori. It is possible to show that a(n) >= 2^(n^2)/n^2 for all n.
Main diagonal of A184271.
Main diagonal of Table 3: The number a(m, n) of toroidal m X n binary arrays, allowing rotation of the rows and/or the columns but not reflection, for m, n = 1, 2, ..., 8, at page 5 of Ethier. - Jonathan Vos Post, Jan 14 2013
This is a 2-dimensional generalization of binary necklaces (A000031). - Gus Wiseman, Feb 04 2019

			From _Gus Wiseman_, Feb 04 2019: (Start)
Inequivalent representatives of the a(2) = 7 checkered tori:
  [0 0] [0 0] [0 0] [0 1] [0 1] [0 1] [1 1]
  [0 0] [0 1] [1 1] [0 1] [1 0] [1 1] [1 1]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..57
S. N. Ethier, Counting toroidal binary arrays, arXiv:1301.2352v1 [math.CO], Jan 10, 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, J. Int. Seq. (2024) Vol. 27, Art. No. 24.6.1. See p. 2.
Wikipedia, Pólya enumeration theorem

Cf. A184271 (n X k toroidal binary arrays).
Cf. A000031, A001037, A008965.
Cf. A323858, A323859, A323861, A323863, A323865, A323870, A323872.

a[n_] := Sum[If[Mod[n, c] == 0, Sum[If[Mod[n, d] == 0, EulerPhi[c] EulerPhi[d] 2^(n^2/LCM[c, d]), 0], {d, 1, n}], 0], {c, 1, n}]/n ^2

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

Terms a(6) and a(7) from A184271
a(8)-a(12) from Stewart N. Ethier, Aug 24 2012
a(0)=1 prepended by Alois P. Heinz, Aug 20 2017

A342527 Number of compositions of n with alternating parts equal.

Original entry on oeis.org

1, 1, 2, 4, 6, 8, 11, 12, 16, 17, 21, 20, 29, 24, 31, 32, 38, 32, 46, 36, 51, 46, 51, 44, 69, 51, 61, 60, 73, 56, 87, 60, 84, 74, 81, 76, 110, 72, 91, 88, 115, 80, 123, 84, 117, 112, 111, 92, 153, 101, 132, 116, 139, 104, 159, 120, 161, 130, 141, 116, 205, 120, 151, 156, 178, 142, 195, 132, 183, 158

Offset: 0

Gus Wiseman, Mar 24 2021

nonn

These are finite sequences q of positive integers summing to n such that q(i) = q(i+2) for all possible i.

			The a(1) = 1 through a(8) = 16 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (12)   (13)    (14)     (15)      (16)       (17)
             (21)   (22)    (23)     (24)      (25)       (26)
             (111)  (31)    (32)     (33)      (34)       (35)
                    (121)   (41)     (42)      (43)       (44)
                    (1111)  (131)    (51)      (52)       (53)
                            (212)    (141)     (61)       (62)
                            (11111)  (222)     (151)      (71)
                                     (1212)    (232)      (161)
                                     (2121)    (313)      (242)
                                     (111111)  (12121)    (323)
                                               (1111111)  (1313)
                                                          (2222)
                                                          (3131)
                                                          (21212)
                                                          (11111111)

Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The odd-length case is A062968.
The even-length case is A065608.
The version with alternating parts unequal is A224958 (unordered: A000726).
The version with alternating parts weakly decreasing is A342528.
A000005 counts constant compositions.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A000203 adds up divisors.
A002843 counts compositions with all adjacent parts x <= 2y.
A003242 counts anti-run compositions.
A175342 counts compositions with constant differences.
A342495 counts compositions with constant first quotients.
A342496 counts partitions with constant first quotients (strict: A342515, ranking: A342522).
Cf. A001522, A008965, A048004, A059966, A064410, A064428, A069916, A114921, A167606, A325545, A325557.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@Plus@@@Reverse/@Partition[#,2,1]&]],{n,0,15}]

a(n) = 1 + n + A000203(n) - 2*A000005(n).

a(n) = A065608(n) + A062968(n).

A345195 Number of non-alternating anti-run compositions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 4, 10, 23, 49, 96, 192, 368, 692, 1299, 2403, 4400, 8029, 14556, 26253, 47206, 84574, 151066, 269244, 478826, 849921, 1506309, 2665829, 4711971, 8319763, 14675786, 25865400, 45552678, 80171353, 141015313, 247905305, 435614270, 765132824

Offset: 0

Gus Wiseman, Jun 17 2021

nonn

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).

An anti-run (separation or Carlitz composition) is a sequence with no adjacent equal parts.

			The a(9) = 23 anti-runs:
  (1,2,6)  (1,2,4,2)  (1,2,1,2,3)
  (1,3,5)  (1,2,5,1)  (1,2,3,1,2)
  (2,3,4)  (1,3,4,1)  (1,2,3,2,1)
  (4,3,2)  (1,4,3,1)  (1,3,2,1,2)
  (5,3,1)  (1,5,2,1)  (2,1,2,3,1)
  (6,2,1)  (2,1,2,4)  (2,1,3,2,1)
           (2,4,2,1)  (3,2,1,2,1)
           (3,1,2,3)
           (3,2,1,3)
           (4,2,1,2)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Non-anti-run compositions are counted by A261983.
A version counting partitions is A345166, ranked by A345173.
These compositions are ranked by A345169.
Non-alternating compositions are counted by A345192, ranked by A345168.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A025047 counts alternating or wiggly compositions, ranked by A345167.
A032020 counts strict compositions.
A106356 counts compositions by number of maximal anti-runs.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A345164 counts alternating permutations of prime indices, w/ twins A344606.
A345165 counts partitions w/o an alternating permutation, ranked by A345171.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A345194 counts alternating patterns (with twins: A344605).
Cf. A005649, A008965, A114901, A178470, A333755, A344604, A344614, A344654, A344740, A345162, A345163, A348380, A348612, A348613.

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
sepQ[y_]:=!MatchQ[y,{_,x_,x_,_}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], sepQ[#]&&!wigQ[#]&]],{n,0,15}]

a(n) = A003242(n) - A025047(n).

a(21) onwards from Andrew Howroyd, Jan 31 2024

A351204 Number of integer partitions of n such that every permutation has all distinct runs.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 8, 9, 11, 14, 18, 20, 25, 28, 34, 41, 47, 53, 64, 72, 84, 98, 113, 128, 148, 169, 194, 223, 255, 289, 333, 377, 428, 488, 554, 629, 715, 807, 913, 1033, 1166, 1313, 1483, 1667, 1875, 2111, 2369, 2655, 2977, 3332, 3729, 4170, 4657, 5195, 5797, 6459

Offset: 0

Gus Wiseman, Feb 15 2022

nonn

Partitions enumerated by this sequence include those in which all parts are either the same or distinct as well as partitions with an even number of parts all of which except one are the same. - Andrew Howroyd, Feb 15 2022

			The a(1) = 1 through a(8) = 11 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (2111)   (51)      (61)       (62)
                            (11111)  (222)     (421)      (71)
                                     (321)     (2221)     (431)
                                     (3111)    (4111)     (521)
                                     (111111)  (211111)   (2222)
                                               (1111111)  (5111)
                                                          (311111)
                                                          (11111111)

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 A000005.
The version for normal multisets is 2^(n-1) - A283353(n-3).
The complement is counted by A351203, ranked by A351201.
A005811 counts runs in binary expansion.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A059966 counts Lyndon compositions, necklaces A008965, aperiodic A000740.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A238130 and A238279 count compositions by number of runs.
A297770 counts distinct runs in binary expansion.
A003242 counts anti-run compositions.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- 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. A000041, A035363, A047993, A116608, A144300, A329746, A351291.

Table[Length[Select[IntegerPartitions[n],Select[Permutations[#],!UnsameQ@@Split[#]&]=={}&]],{n,0,15}]

\\ here Q(n) is A000009.
Q(n)={polcoef(prod(k=1, n, 1 + x^k + O(x*x^n)), n)}
a(n)={Q(n) + if(n, numdiv(n) - 1) + sum(k=1, (n-1)\3, sum(j=3, (n-1)\k, j%2==1 && n-k*j<>k))} \\ Andrew Howroyd, Feb 15 2022

Terms a(26) and beyond from Andrew Howroyd, Feb 15 2022

A000939 Number of inequivalent n-gons.

Original entry on oeis.org

1, 1, 1, 2, 4, 14, 54, 332, 2246, 18264, 164950, 1664354, 18423144, 222406776, 2905943328, 40865005494, 615376173184, 9880209206458, 168483518571798, 3041127561315224, 57926238289970076, 1161157777643184900, 24434798429947993054, 538583682082245127336

Offset: 1

N. J. A. Sloane

Here two n-gons are said to be equivalent if they differ in starting point, orientation, or by a rotation (but not by a reflection - for that see A000940).

Number of cycle necklaces on n vertices, defined as equivalence classes of (labeled, undirected) Hamiltonian cycles under rotation of the vertices. The path version is A275527. - Gus Wiseman, Mar 02 2019

			Possibilities for n-gons without distinguished vertex can be encoded as permutation classes of vertices, two permutations being equivalent if they can be obtained from each other by circular rotation, translation mod n or complement to n+1.
n=3: 123.
n=4: 1234, 1243.
n=5: 12345, 12354, 12453, 13524.
n=6: 123456, 123465, 123564, 123645, 123654, 124365, 124635, 124653, 125364, 125463, 125634, 126435, 126453, 135264.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Georg Fischer, Table of n, a(n) for n = 1..450 (first 100 terms by T. D. Noe)
S. W. Golomb and L. R. Welch, On the enumeration of polygons, Amer. Math. Monthly, 67 (1960), 349-353.
S. W. Golomb and L. R. Welch, On the enumeration of polygons, Amer. Math. Monthly, 67 (1960), 349-353. [Annotated scanned copy]
Samuel Herman and Eirini Poimenidou, Orbits of Hamiltonian Paths and Cycles in Complete Graphs, arXiv:1905.04785 [math.CO], 2019.
A. Stoimenow, Enumeration of chord diagrams and an upper bound for Vassiliev invariants, J. Knot Theory Ramifications, 7 (1998), no. 1, 93-114.
Eric Weisstein's World of Mathematics, Hamiltonian Cycle.
Wikipedia, Hamiltonian path.
Wikipedia, Polygon.
Gus Wiseman, Inequivalent representatives of the a(6) = 14 cycle necklaces.
Gus Wiseman, Inequivalent representatives of the a(7) = 54 n-gons.

Cf. A000940. Bisections give A094154, A094155.
For star polygons see A231091.
Cf. A000031, A002619, A002866, A006125, A008965, A059966, A060223, A192332, A275527, A323858, A323870, A324461.

with(numtheory):
# for n odd:
Ed:= proc(n) local t1, d; t1:=0; for d from 1 to n do
       if n mod d = 0 then t1:= t1+phi(n/d)^2*d!*(n/d)^d fi od:
       t1/(2*n^2)
     end:
# for n even:
Ee:= proc(n) local t1, d; t1:= 2^(n/2)*(n/2)*(n/2)!; for d
       from 1 to n do if n mod d = 0 then t1:= t1+
       phi(n/d)^2*d!*(n/d)^d; fi od: t1/(2*n^2)
     end:
A000939:= n-> if n mod 2 = 0 then ceil(Ee(n)) else ceil(Ed(n)); fi:
seq(A000939(n), n=1..25);

a[n_] := (t = If[OddQ[n], 0, 2^(n/2)*(n/2)*(n/2)!]; Do[If[Mod[n, d]==0, t = t+EulerPhi[n/d]^2*d!*(n/d)^d], {d, 1, n}]; t/(2*n^2)); a[1] := 1; a[2] := 1; Print[a /@ Range[1, 450]] (* Jean-François Alcover, May 19 2011, after Maple prog. *)
rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
Table[Length[Select[Union[Sort[Sort/@Partition[#,2,1,1]]&/@Permutations[Range[n]]],#==First[Sort[Table[Nest[rotgra[#,n]&,#,j],{j,n}]]]&]],{n,8}] (* Gus Wiseman, Mar 02 2019 *)

a(n)={if(n<3, n>=0, (if(n%2, 0, (n/2-1)!*2^(n/2-2)) + sumdiv(n, d, eulerphi(n/d)^2 * d! * (n/d)^d)/n^2)/2)} \\ Andrew Howroyd, Aug 17 2019

For formula see Maple lines.
a(2*n + 1) =  A002619(2*n + 1)/2 for n > 0; a(2*n) = (A002619(2*n) + A002866(n-1))/2 for n > 1. - Andrew Howroyd, Aug 17 2019
a(n) ~ sqrt(2*Pi)/2 * n^(n-3/2) / e^n. - Ludovic Schwob, Nov 03 2022

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 05 2004

Added a(1) = 1 and a(2) = 1 by Gus Wiseman, Mar 02 2019

cp's OEIS Frontend

A329745 Number of compositions of n with runs-resistance 2.

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A342192 Heinz numbers of partitions of crank 0.

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A329766 Number of compositions of n whose run-lengths cover an initial interval of positive integers.

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A333943 Numbers k such that the k-th composition in standard order is a reversed necklace.

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A037306 Triangle T(n,k) read by rows: the number of compositions of n into k parts, modulo cyclic shifts.

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A179043 Number of n X n checkered tori.

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A342527 Number of compositions of n with alternating parts equal.

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