A106351 -id:A106351 - cp's OEIS frontend

A335432 Number of anti-run permutations of the prime indices of Mersenne numbers A000225(n) = 2^n - 1.

1, 1, 1, 2, 1, 1, 1, 6, 2, 6, 2, 36, 1, 6, 6, 24, 1, 24, 1, 240, 6, 24, 2, 1800, 6, 6, 6, 720, 6, 1800, 1, 120, 24, 6, 24, 282240, 2, 6, 24, 15120, 2, 5760, 6, 5040, 720, 24, 6, 1451520, 2, 5040, 120, 5040, 6, 1800, 720, 40320, 24, 720, 2, 1117670400, 1, 6, 1800, 5040, 6

Offset: 1

Gus Wiseman, Jul 02 2020

nonn

An anti-run is a sequence with no adjacent equal parts.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(1) = 1 through a(10) = 6 permutations:
  ()  (2)  (4)  (2,3)  (11)  (2,4,2)  (31)  (2,3,7)  (21,4)  (11,2,5)
                (3,2)                       (2,7,3)  (4,21)  (11,5,2)
                                            (3,2,7)          (2,11,5)
                                            (3,7,2)          (2,5,11)
                                            (7,2,3)          (5,11,2)
                                            (7,3,2)          (5,2,11)

Andrew Howroyd, Table of n, a(n) for n = 1..200

The version for factorial numbers is A335407.
Anti-run compositions are A003242.
Anti-run patterns are A005649.
Permutations of prime indices are A008480.
Anti-runs are ranked by A333489.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Strict permutations of prime indices are A335489.
Cf. A056239, A106351, A112798, A114938, A278990, A292884, A325535, A335125.
Mersenne numbers: A000225, A046051, A046800, A046801, A049093, A059305, A159611, A325610, A325611, A325612, A325625.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[2^n-1]],!MatchQ[#,{_,x_,x_,_}]&]],{n,0,30}]

\\ See A335452 for count.
a(n) = {count(factor(2^n-1)[,2])} \\ Andrew Howroyd, Feb 03 2021

a(n) = A335452(A000225(n)).

Terms a(51) and beyond from Andrew Howroyd, Feb 03 2021

A335461 Triangle read by rows where T(n,k) is the number of patterns of length n with k runs.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 4, 8, 0, 1, 6, 24, 44, 0, 1, 8, 48, 176, 308, 0, 1, 10, 80, 440, 1540, 2612, 0, 1, 12, 120, 880, 4620, 15672, 25988, 0, 1, 14, 168, 1540, 10780, 54852, 181916, 296564, 0, 1, 16, 224, 2464, 21560, 146272, 727664, 2372512, 3816548

Offset: 0

Gus Wiseman, Jul 03 2020

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.

			Triangle begins:
     1
     0     1
     0     1     2
     0     1     4     8
     0     1     6    24    44
     0     1     8    48   176   308
     0     1    10    80   440  1540  2612
     0     1    12   120   880  4620 15672 25988
Row n = 3 counts the following patterns:
  (1,1,1)  (1,1,2)  (1,2,1)
           (1,2,2)  (1,2,3)
           (2,1,1)  (1,3,2)
           (2,2,1)  (2,1,2)
                    (2,1,3)
                    (2,3,1)
                    (3,1,2)
                    (3,2,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A000670.
Column n = k is A005649 (anti-run patterns).
Central coefficients are A337564.
The version for compositions is A333755.
Runs of standard compositions are counted by A124767.
Run-lengths of standard compositions are A333769.
Cf. A003242, A052841, A060223, A106351, A106356, A269134, A325535, A333489, A333627, A335838.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],Length[Split[#]]==k&]],{n,0,5},{k,0,n}]

\\ here b(n) is A005649.
b(n) = {sum(k=0, n, stirling(n,k,2)*(k + 1)!)}
T(n,k)=if(n==0, k==0, b(k-1)*binomial(n-1,k-1)) \\ Andrew Howroyd, Dec 31 2020

T(n,k) = A005649(k-1) * binomial(n-1,k-1) for k > 0. - Andrew Howroyd, Dec 31 2020

A337504 Number of compositions of 2*n with n maximal anti-runs.

Original entry on oeis.org

1, 1, 3, 8, 13, 33, 112, 286, 769, 2288, 6695, 18745, 54654, 160888, 467402, 1362378, 4016517, 11807966, 34708018, 102451390, 302870005, 895207191, 2650590597, 7859253320, 23316653154, 69231883374, 205773157904, 612021943421, 1821435719846, 5424528040529, 16165017705176

Offset: 0

Gus Wiseman, Sep 04 2020

nonn

An anti-run is a sequence with no adjacent equal parts.

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

Andrew Howroyd, Table of n, a(n) for n = 0..200

A106356 has this as main diagonal n = 2*k.
A336108 is the version for runs.
A337505 is the version for patterns.
A337564 is the version for runs in patterns.
A003242 counts anti-run compositions.
A011782 counts compositions.
A124767 counts runs in standard compositions.
A238343 counts compositions by descents.
A333213 counts compositions by weak ascents.
A333381 counts anti-runs in standard compositions.
A333382 counts adjacent unequal pairs in standard compositions.
A333489 ranks anti-runs.
A333755 counts compositions by number of runs.
A333769 gives run-lengths in standard compositions.
A337565 gives anti-run lengths in standard compositions.
Cf. A106351, A124762, A233564, A235998, A238130, A238279, A333214, A333216.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[2*n],Length[Split[#,UnsameQ]]==n&]],{n,0,10}]

a(n)={polcoef(polcoef(1 - y + y*(y-1)/(y - 1 - sum(d=1, 2*n, (y-1)^d*x^d/(1 - x^d) + O(x^(2*n+1)))), 2*n, x), n, y)} \\ Andrew Howroyd, Feb 02 2021

a(n) = [x^(2*n)*y^n] 1 - y + y*(y-1)/(y - 1 - Sum_{d>=1} (y-1)^d*x^d/(1 - x^d)). - Andrew Howroyd, Feb 02 2021

Terms a(11) and beyond from Andrew Howroyd, Feb 02 2021

A386576 Number of anti-runs of length n covering an initial interval of positive integers with strictly decreasing multiplicities.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 10, 4, 14, 84, 1136, 967, 3342, 12823, 101762, 1769580

Offset: 0

Gus Wiseman, Aug 03 2025

An anti-run is a sequence with no adjacent equal terms.

			The a(7) = 4 anti-runs are:
  (1,2,1,2,1,2,1)
  (1,2,1,2,1,3,1)
  (1,2,1,3,1,2,1)
  (1,3,1,2,1,2,1)

For any multiplicities we have A005649.
For weakly instead of strictly decreasing multiplicities we have A321688.
A003242 and A335452 count anti-runs, ranks A333489.
A005651 counts ordered set partitions with weakly decreasing sizes, strict A007837.
A032020 counts strict anti-run compositions.
A325534 counts separable multisets, ranks A335433.
A325535 counts inseparable multisets, ranks A335448.
A336103 counts normal separable multisets, inseparable A336102.
A386583 counts separable partitions by length, inseparable A386584.
A386585 counts partitions of separable type by length, sums A336106, ranks A335127.
A386586 counts partitions of inseparable type by length, sums A025065, ranks A335126.
A386633 counts separable set partitions, row sums of A386635.
A386634 counts inseparable set partitions, row sums of A386636.
Cf. A000670, A019472, A106351, A238130, A335125, A335516, A386580, A386581.

seps[ptn_,fir_]:=If[Total[ptn]==1,{{fir}},Join@@Table[Prepend[#,fir]&/@seps[MapAt[#-1&,ptn,fir],nex],{nex,Select[DeleteCases[Range[Length[ptn]],fir],ptn[[#]]>0&]}]];
seps[ptn_]:=If[Total[ptn]==0,{{}},Join@@(seps[ptn,#]&/@Range[Length[ptn]])];
Table[Sum[Length[seps[y]],{y,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,10}]

A386578 Irregular triangle read by rows where T(n,k) is the number of permutations of row n of A305936 (a multiset whose multiplicities are the prime indices of n) with k adjacent equal parts.

Original entry on oeis.org

1, 0, 1, 2, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 1, 6, 0, 0, 2, 2, 2, 0, 0, 2, 2, 0, 0, 0, 0, 0, 1, 6, 6, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 3, 2, 0, 1, 4, 3, 2, 0, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 12, 12, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 12, 6, 0, 0, 0, 3, 6, 4, 2, 0

Offset: 2

Gus Wiseman, Aug 04 2025

Row 1 is empty, so offset is 2.
Same as A386579 with rows reversed.
This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			Row n = 21 counts the following permutations:
  .  112121  111212  111221  111122  .
     121121  112112  112211  221111
     121211  121112  122111
             211121  211112
             211211
             212111
Triangle begins
   .
   1
   0  1
   2  0
   0  0  1
   1  2  0
   0  0  0  1
   6  0  0
   2  2  2  0
   0  2  2  0
   0  0  0  0  1
   6  6  0  0
   0  0  0  0  0  1
   0  0  3  2  0
   1  4  3  2  0
  24  0  0  0
   0  0  0  0  0  0  1
  12 12  6  0  0
   0  0  0  0  0  0  0  1
   2 12  6  0  0
   0  3  6  4  2  0

Column k = last is A010051.
Row lengths are A056239.
Initial zeros are counted by A252736 = A001222 - 1.
Row sums are A318762.
Column k = 0 is A335125.
For prime indices we have A386577.
Reversing all rows gives A386579.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A025065(n - 2) counts partitions of inseparable type, ranks A335126, sums of A386586.
A124762 gives inseparability of standard compositions, separability A333382.
A305936 is a multiset whose multiplicities are the prime indices of n.
A325534 counts separable multisets, ranks A335433, sums of A386583.
A325535 counts inseparable multisets, ranks A335448, sums of A386584.
A336106 counts partitions of separable type, ranks A335127, sums of A386585.
Cf. A001221, A003963, A051903, A051904, A106351, A112798, A238130, A336102, A373957, A374246, A382525, A386581.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
aqt[c_,x_]:=Select[Permutations[c],Function[q,Length[Select[Range[Length[q]-1],q[[#]]==q[[#+1]]&]]==x]];
Table[Table[Length[aqt[nrmptn[n],k]],{k,0,Length[nrmptn[n]]-1}],{n,30}]

A221235 Number of nonnegative integer arrays of length n summing to n without adjacent equal values.

Original entry on oeis.org

1, 1, 2, 7, 14, 30, 76, 188, 444, 1075, 2656, 6504, 15926, 39316, 97252, 240597, 596686, 1482971, 3689768, 9191590, 22927718, 57253334, 143101896, 358003469, 896391914, 2246149936, 5632310800, 14132640565, 35483595966, 89141668532, 224061612932, 563473301874

Offset: 0

R. H. Hardin, Jan 10 2013

nonn

			All solutions for n=3
..1....2....0....2....0....0....1
..2....1....3....0....2....1....0
..0....0....0....1....1....2....2

Alois P. Heinz, Table of n, a(n) for n = 0..500 (first 210 terms from R. H. Hardin)

Cf. A106351.

b:= proc(n, h, t) option remember; `if`(t<2, `if`(n<>h, 1, 0),
      add(`if`(h=j, 0, b(n-j, `if`(j>n-j, -1, j), t-1)), j=0..n))
    end:
a:= n-> b(n, -1, n):
seq(a(n), n=0..35);  # Alois P. Heinz, Oct 15 2017

b[n_, h_, t_] := b[n, h, t] = If[t < 2, If[n != h, 1, 0],
     Sum[If[h == j, 0, b[n-j, If[j > n-j, -1, j], t-1]], {j, 0, n}]];
a[n_] := b[n, -1, n];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Apr 28 2022, after Alois P. Heinz *)

a(n) = A106351(2n,n). - Alois P. Heinz, Oct 12 2017

a(0)=1 prepended by Alois P. Heinz, Oct 15 2017

A293595 Triangle read by rows: T(n,k) = number of compositions of n into k parts such that no two cyclically adjacent parts are equal.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 2, 0, 0, 1, 4, 0, 0, 0, 1, 4, 6, 2, 0, 0, 1, 6, 6, 4, 0, 0, 0, 1, 6, 12, 10, 0, 0, 0, 0, 1, 8, 18, 16, 10, 2, 0, 0, 0, 1, 8, 24, 40, 20, 6, 0, 0, 0, 0, 1, 10, 30, 52, 50, 18, 0, 0, 0, 0, 0, 1, 10, 42, 84, 90, 50, 14, 2, 0, 0, 0, 0

Offset: 1

Andrew Howroyd, Oct 12 2017

Compositions of length 1 are included.

See theorem 4 in Hadjicostas reference for generating function.

			Triangle begins:
  1;
  1,  0;
  1,  2,  0;
  1,  2,  0,  0;
  1,  4,  0,  0,  0;
  1,  4,  6,  2,  0,  0;
  1,  6,  6,  4,  0,  0,  0;
  1,  6, 12, 10,  0,  0,  0,  0;
  1,  8, 18, 16, 10,  2,  0,  0,  0;
  1,  8, 24, 40, 20,  6,  0,  0,  0,  0;
  ...
Case n=6:
The included compositions are:
k=1: 6;                                => T(6,1) = 1
k=2: 15, 24, 42, 51;                   => T(6,2) = 4
k=3: 123, 132, 213, 231, 312, 321;     => T(6,3) = 6
k=4: 1212, 2121;                       => T(6,4) = 2

Andrew Howroyd, Table of n, a(n) for n = 1..1275
P. Hadjicostas, Cyclic, dihedral and symmetric Carlitz compositions of a positive integer, Journal of Integer Sequences, 20 (2017), Article 17.8.5.

Row sums are in A212322.

Cf. A106351, A106369.

max = 10; gf = Sum[x^(2*j)*y^2/(1 + x^j*y), {j, 1, max}] + Sum[x^j*y/(1 + x^j*y)^2, {j, 1, max}]/(1 - Sum[ x^j*y/(1 + x^j*y), {j, 1, max}]) + O[x]^(max+1) + O[y]^(max+1) // Normal // Expand;
T[n_, k_] := SeriesCoefficient[gf, {x, 0, n}, {y, 0, k}];
Table[T[n, k], {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 19 2018 *)

gf(n,y) = {my(A=sum(j=1, n, x^(2*j)*y^2/(1+x^j*y) + O(x*x^n)),
B=sum(j=1, n, x^j*y/(1+x^j*y)^2 + O(x*x^n)),
C=sum(j=1, n, x^j*y/(1+x^j*y) + O(x*x^n)));
A + B/(1-C)}
for(n=1,10,my(p=polcoeff(gf(n,y),n));for(k=1,n,print1(polcoeff(p,k),", "));print)

G.f.: (Sum_{j>=1} x^(2*j)*y^2/(1+x^j*y)) + (Sum_{j>=1} x^j*y/(1+x^j*y)^2) / (1 - Sum_{j>=1} x^j*y/(1+x^j*y)).

A335459 Number of permutations of the prime indices of n! with at least one non-singleton run.

Original entry on oeis.org

0, 0, 0, 0, 4, 18, 102, 786, 3960, 51450, 675570, 10804710, 139674024, 2793377664, 58662908640, 1798893694080, 26985313555200, 782574083010720, 25992638958686400, 857757034323189000, 30021498596590300800, 1563341714743040232000, 64179292280096037844800, 2631350957341279888915200

Offset: 0

Gus Wiseman, Jul 03 2020

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(4) = 4 and a(5) = 18 permutations:
  (1,1,1,2)  (1,1,1,2,3)
  (1,1,2,1)  (1,1,1,3,2)
  (1,2,1,1)  (1,1,2,1,3)
  (2,1,1,1)  (1,1,2,3,1)
             (1,1,3,1,2)
             (1,1,3,2,1)
             (1,2,1,1,3)
             (1,2,3,1,1)
             (1,3,1,1,2)
             (1,3,2,1,1)
             (2,1,1,1,3)
             (2,1,1,3,1)
             (2,1,3,1,1)
             (2,3,1,1,1)
             (3,1,1,1,2)
             (3,1,1,2,1)
             (3,1,2,1,1)
             (3,2,1,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..200

The anti-run version is A335407.
Anti-runs are ranked by A333489.
Anti-run compositions are A003242.
Anti-run patterns are A005649.
Permutations of prime indices are A008480.
Permutations of prime indices of n! are A325617.
Anti-run permutations of prime indices are A335452.
Cf. A001222, A022559, A056239, A106351, A112798, A114938, A325535, A335125.
Factorial numbers: A000142, A002982, A007489, A027423, A054991, A108731, A325272, A325273, A325617.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n!]],MatchQ[#,{_,x_,x_,_}]&]],{n,0,10}]

\\ See A335452 for count.
a(n)={my(sig=factor(n!)[, 2]); vecsum(sig)!/vecprod([k! | k<-sig]) - count(sig)} \\ Andrew Howroyd, Apr 17 2021

A008480(n!) = a(n) + A335407(n).

a(11)-a(13) from Vaclav Kotesovec, Jul 07 2020

Terms a(14) and beyond from Andrew Howroyd, Apr 17 2021

A337506 Triangle read by rows where T(n,k) is the number of length-n sequences covering an initial interval of positive integers with k maximal anti-runs.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 8, 4, 1, 0, 44, 24, 6, 1, 0, 308, 176, 48, 8, 1, 0, 2612, 1540, 440, 80, 10, 1, 0, 25988, 15672, 4620, 880, 120, 12, 1, 0, 296564, 181916, 54852, 10780, 1540, 168, 14, 1, 0, 3816548, 2372512, 727664, 146272, 21560, 2464, 224, 16, 1

Offset: 0

Gus Wiseman, Sep 06 2020

An anti-run is a sequence with no adjacent equal parts. The number of maximal anti-runs is one more than the number of adjacent equal parts.

			Triangle begins:
  1
  0      1
  0      2      1
  0      8      4      1
  0     44     24      6      1
  0    308    176     48      8      1
  0   2612   1540    440     80     10      1
  0  25988  15672   4620    880    120     12      1
  0 296564 181916  54852  10780   1540    168     14      1
Row n = 3 counts the following sequences (empty column indicated by dot):
  .  (1,2,1)  (1,1,2)  (1,1,1)
     (1,2,3)  (1,2,2)
     (1,3,2)  (2,1,1)
     (2,1,2)  (2,2,1)
     (2,1,3)
     (2,3,1)
     (3,1,2)
     (3,2,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1325

A000670 gives row sums.
A005649 gives column k = 1.
A337507 gives column k = 2.
A337505 gives the diagonal n = 2*k.
A106356 is the version for compositions.
A238130/A238279/A333755 is the version for runs in compositions.
A335461 has the reversed rows (except zeros).
A003242 counts anti-run compositions.
A124762 counts adjacent equal terms in standard compositions.
A124767 counts maximal runs in standard compositions.
A333381 counts maximal anti-runs in standard compositions.
A333382 counts adjacent unequal terms in standard compositions.
A333489 ranks anti-run compositions.
A333769 gives maximal run-lengths in standard compositions.
A337565 gives maximal anti-run lengths in standard compositions.
Cf. A019472, A052841, A060223, A106351, A269134, A325535, A337564.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],Length[Split[#,UnsameQ]]==k&]],{n,0,5},{k,0,n}]

\\ here b(n) is A005649.
b(n) = {sum(k=0, n, stirling(n,k,2)*(k + 1)!)}
T(n,k)=if(n==0, k==0, b(n-k)*binomial(n-1,k-1)) \\ Andrew Howroyd, Dec 31 2020

T(n,k) = A005649(n-k) * binomial(n-1,k-1) for k > 0. - Andrew Howroyd, Dec 31 2020

Terms a(45) and beyond from Andrew Howroyd, Dec 31 2020

A106355 Number of compositions of n into 6 parts such that no two adjacent parts are equal.

Original entry on oeis.org

2, 10, 30, 76, 168, 320, 580, 968, 1558, 2380, 3540, 5078, 7160, 9804, 13238, 17510, 22884, 29418, 37462, 47054, 58638, 72272, 88454, 107262, 129312, 154644, 183994, 217442, 255782, 299114, 348386, 403652, 466012, 535550, 613442, 699812, 796012

Offset: 9

Christian G. Bower, Apr 29 2005

Alois P. Heinz, Table of n, a(n) for n = 9..1000
A. Knopfmacher and H. Prodinger, On Carlitz compositions, European Journal of Combinatorics, Vol. 19, 1998, pp. 579-589.

Column 6 of A106351. Cf. A003242.

G.f.: 2 *(16*x^12 +16*x^11 +31*x^10 +40*x^9 +53*x^8 +51*x^7 +51*x^6 +39*x^5 +31*x^4 +18*x^3 +9*x^2 +4*x+1) *x^9 / ((x^2-x+1) *(x^4+x^3+x^2+x+1) *(x^2+1) *(x^2+x+1)^2 *(x+1)^3 *(x-1)^6). - Alois P. Heinz, Sep 04 2015

Replaced broken link, Vaclav Kotesovec, May 01 2014

cp's OEIS Frontend

A335432 Number of anti-run permutations of the prime indices of Mersenne numbers A000225(n) = 2^n - 1.

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A335461 Triangle read by rows where T(n,k) is the number of patterns of length n with k runs.

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A337504 Number of compositions of 2*n with n maximal anti-runs.

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A386576 Number of anti-runs of length n covering an initial interval of positive integers with strictly decreasing multiplicities.

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A386578 Irregular triangle read by rows where T(n,k) is the number of permutations of row n of A305936 (a multiset whose multiplicities are the prime indices of n) with k adjacent equal parts.

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A221235 Number of nonnegative integer arrays of length n summing to n without adjacent equal values.

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Maple

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A293595 Triangle read by rows: T(n,k) = number of compositions of n into k parts such that no two cyclically adjacent parts are equal.

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