A269134 -id:A269134 - cp's OEIS frontend

A316222 Number of positive subset-sum triangles whose composite is a positive subset-sum of an integer partition of n.

1, 5, 20, 74, 258, 855, 2736, 8447

Offset: 1

Gus Wiseman, Jun 27 2018

A positive subset-sum is a pair (h,g), where h is a positive integer and g is an integer partition, such that some submultiset of g sums to h. A triangle consists of a root sum r and a sequence of positive subset-sums ((h_1,g_1),...,(h_k,g_k)) such that the sequence (h_1,...,h_k) is weakly decreasing and has a submultiset summing to r.

			We write positive subset-sum triangles in the form rootsum(branch,...,branch). The a(2) = 5 positive subset-sum triangles:
  2(2(2))
  1(1(1,1))
  2(2(1,1))
  1(1(1),1(1))
  2(1(1),1(1))

Cf. A063834, A262671, A269134, A276024, A281113, A301934, A301935, A316219, A316220.

A318567 Number of pairs (c, y) where c is an integer composition and y is an integer partition and y can be obtained from c by choosing a partition of each part, flattening, and sorting.

Original entry on oeis.org

1, 3, 8, 21, 54, 137, 343, 847, 2075, 5031, 12109, 28921, 68633, 161865, 379655

Offset: 1

Gus Wiseman, Aug 29 2018

Also the number of combinatory separations of normal multisets of weight n with constant parts. A multiset is normal if it spans an initial interval of positive integers. The type of a multiset is the unique normal multiset that has the same sequence of multiplicities when its entries are taken in increasing order. For example the type of 335556 is 112223. A pair h<={g_1,...,g_k} is a combinatory separation iff there exists a multiset partition of h whose multiset of block-types is {g_1,...,g_k}.

			The a(3) = 8 combinatory separations:
  111<={111}
  111<={1,11}
  111<={1,1,1}
  112<={1,11}
  112<={1,1,1}
  122<={1,11}
  122<={1,1,1}
  123<={1,1,1}

Cf. A000041, A007716, A011782, A034691, A255906, A265947, A269134.

Cf. A317533, A317791, A318396, A318559, A318560, A318562, A318565.

Table[Sum[Length[Union[Sort/@Join@@@Tuples[IntegerPartitions/@c]]],{c,Join@@Permutations/@IntegerPartitions[n]}],{n,30}]

A330472 Triangle read by rows where T(n,k) is the number of non-isomorphic k-element multisets of nonempty multisets of nonempty multisets (all finite).

Original entry on oeis.org

1, 0, 1, 0, 4, 2, 0, 10, 8, 3, 0, 33, 48, 18, 5, 0, 91, 204, 118, 32, 7, 0, 298, 959, 743, 266, 58, 11, 0, 910, 4193, 4334, 1927, 519, 94, 15, 0, 3017, 18947, 25305, 13992, 4407, 966, 154, 22, 0, 9945, 84798, 145033, 97947, 36410, 9023, 1679, 236, 30

Offset: 0

Gus Wiseman, Dec 19 2019

			Triangle begins:
   1
   0   1
   0   4   2
   0  10   8   3
   0  33  48  18   5
   0  91 204 118  32   7
   0 298 959 743 266  58  11
For example, row n = 3 counts the following multiset partitions:
  {{111}}      {{1}}{{11}}    {{1}}{{1}}{{1}}
  {{112}}      {{1}}{{12}}    {{1}}{{1}}{{2}}
  {{123}}      {{1}}{{23}}    {{1}}{{2}}{{3}}
  {{1}{11}}    {{2}}{{11}}
  {{1}{12}}    {{1}}{{1}{1}}
  {{1}{23}}    {{1}}{{1}{2}}
  {{2}{11}}    {{1}}{{2}{3}}
  {{1}{1}{1}}  {{2}}{{1}{1}}
  {{1}{1}{2}}
  {{1}{2}{3}}

Andrew Howroyd, Table of n, a(n) for n = 0..350

Row sums are A318566.
Column k = 1 is A007716 (for n > 0).
Column k = n is A000041.
Partitions of partitions of partitions are A007713.
Twice-factorizations are A050336.
If this is the 3-dimensional version, the 2-dimensional version is A317533.
See A330473 for a variation.
Cf. A001055, A050336, A061260, A269134, A292504, A306186, A317791.

\\ See links in A339645 for combinatorial species functions.
ColGf(k,n)={my(A=symGroupSeries(n)); OgfSeries(sCartProd(sExp(A), sSubstOp(polcoef(A,k,x)*x^k + O(x*x^n), sExp(A)) ))}
M(n,m=n)={Mat(vector(m+1, k, Col(ColGf(k-1,n), -(n+1))))}
{ my(A=M(10)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Jan 17 2023

Terms a(21) and beyond from Andrew Howroyd, Jan 17 2023

A335476 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,1,2).

Original entry on oeis.org

14, 28, 29, 30, 46, 54, 56, 57, 58, 59, 60, 61, 62, 78, 84, 92, 93, 94, 102, 108, 109, 110, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 142, 156, 157, 158, 168, 169, 172, 174, 180, 182, 184, 185, 186, 187, 188, 189, 190, 198, 204

Offset: 1

Gus Wiseman, Jun 18 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. 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.

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. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The sequence of terms together with the corresponding compositions begins:
  14: (1,1,2)
  28: (1,1,3)
  29: (1,1,2,1)
  30: (1,1,1,2)
  46: (2,1,1,2)
  54: (1,2,1,2)
  56: (1,1,4)
  57: (1,1,3,1)
  58: (1,1,2,2)
  59: (1,1,2,1,1)
  60: (1,1,1,3)
  61: (1,1,1,2,1)
  62: (1,1,1,1,2)
  78: (3,1,1,2)
  84: (2,2,3)

Wikipedia, Permutation pattern
Gus Wiseman, Statistics, classes, and transformations of standard compositions
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The complement A335522 is the avoiding version.
The (2,1,1)-matching version is A335478.
Patterns matching this pattern are counted by A335509 (by length).
Permutations of prime indices matching this pattern are counted by A335446.
These compositions are counted by A335470 (by sum).
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335446, A335456, A335458.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],MatchQ[stc[#],{_,x_,_,x_,_,y_,_}/;x

A335477 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (2,2,1).

Original entry on oeis.org

21, 43, 45, 53, 73, 85, 86, 87, 91, 93, 107, 109, 117, 146, 147, 149, 153, 165, 169, 171, 172, 173, 174, 175, 181, 182, 183, 187, 189, 201, 213, 214, 215, 219, 221, 235, 237, 245, 273, 277, 293, 294, 295, 297, 299, 301, 306, 307, 309, 313, 325, 329, 331, 333

Offset: 1

Gus Wiseman, Jun 18 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. 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.

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. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The sequence of terms together with the corresponding compositions begins:
   21: (2,2,1)
   43: (2,2,1,1)
   45: (2,1,2,1)
   53: (1,2,2,1)
   73: (3,3,1)
   85: (2,2,2,1)
   86: (2,2,1,2)
   87: (2,2,1,1,1)
   91: (2,1,2,1,1)
   93: (2,1,1,2,1)
  107: (1,2,2,1,1)
  109: (1,2,1,2,1)
  117: (1,1,2,2,1)
  146: (3,3,2)
  147: (3,3,1,1)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

The complement A335524 is the avoiding version.
The (1,2,2)-matching version is A335475.
Patterns matching this pattern are counted by A335509 (by length).
Permutations of prime indices matching this pattern are counted by A335453.
These compositions are counted by A335472 (by sum).
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335446, A335456, A335458.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],MatchQ[stc[#],{_,x_,_,x_,_,y_,_}/;x>y]&]

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

A318816 Regular tetrangle where T(n,k,i) is the number of non-isomorphic multiset partitions of length i of multiset partitions of length k of multisets of size n.

Original entry on oeis.org

1, 2, 2, 2, 3, 4, 4, 3, 4, 3, 5, 14, 14, 9, 20, 9, 5, 14, 9, 5, 7, 28, 28, 33, 80, 33, 16, 68, 52, 16, 7, 28, 33, 16, 7, 11, 69, 69, 104, 266, 104, 74, 356, 282, 74, 29, 199, 253, 118, 29, 11, 69, 104, 74, 29, 11, 15, 134, 134, 294, 800, 294, 263, 1427, 1164

Offset: 1

Gus Wiseman, Sep 04 2018

			Tetrangle begins:
  1   2     3        5             7
      2 2   4 4     14 14         28 28
            3 4 3    9 20  9      33 80 33
                     5 14  9  5   16 68 52 16
                                   7 28 33 16  7
Non-isomorphic representatives of the T(4,3,2) = 20 multiset partitions:
  {{{1}},{{1},{1,1}}}  {{{1,1}},{{1},{1}}}
  {{{1}},{{1},{1,2}}}  {{{1,1}},{{1},{2}}}
  {{{1}},{{1},{2,2}}}  {{{1,1}},{{2},{2}}}
  {{{1}},{{1},{2,3}}}  {{{1,1}},{{2},{3}}}
  {{{1}},{{2},{1,1}}}  {{{1,2}},{{1},{1}}}
  {{{1}},{{2},{1,2}}}  {{{1,2}},{{1},{2}}}
  {{{1}},{{2},{1,3}}}  {{{1,2}},{{1},{3}}}
  {{{1}},{{2},{3,4}}}  {{{1,2}},{{3},{4}}}
  {{{2}},{{1},{1,1}}}  {{{2,3}},{{1},{1}}}
  {{{2}},{{1},{1,3}}}
  {{{2}},{{3},{1,1}}}

Cf. A007716, A050336, A050338, A255906, A269134, A317533, A317791, A318393, A318399, A318564, A318565, A318566.

A330473 Regular triangle where T(n,k) is the number of non-isomorphic multiset partitions of k-element multiset partitions of multisets of size n.

Original entry on oeis.org

1, 0, 1, 0, 2, 4, 0, 3, 8, 10, 0, 5, 28, 38, 33, 0, 7, 56, 146, 152, 91, 0, 11, 138, 474, 786, 628, 298, 0, 15, 268, 1388, 3117, 3808, 2486, 910, 0, 22, 570, 3843, 11830, 19147, 18395, 9986, 3017, 0, 30, 1072, 10094, 40438, 87081, 110164, 86388, 39889, 9945

Offset: 0

Gus Wiseman, Dec 20 2019

As an alternative description, T(n,k) is the number of non-isomorphic multisets of nonempty multisets of nonempty multisets with n leaves whose multiset union consists of k multisets.

			Triangle begins:
   1
   0   1
   0   2   4
   0   3   8  10
   0   5  28  38  33
   0   7  56 146 152  91
   0  11 138 474 786 628 298
For example, row n = 3 counts the following multiset partitions:
  {{111}}  {{1}{11}}    {{1}{1}{1}}
  {{112}}  {{1}{12}}    {{1}{1}{2}}
  {{123}}  {{1}{23}}    {{1}{2}{3}}
           {{2}{11}}    {{1}}{{1}{1}}
           {{1}}{{11}}  {{1}}{{1}{2}}
           {{1}}{{12}}  {{1}}{{2}{3}}
           {{1}}{{23}}  {{2}}{{1}{1}}
           {{2}}{{11}}  {{1}}{{1}}{{1}}
                        {{1}}{{1}}{{2}}
                        {{1}}{{2}}{{3}}

Andrew Howroyd, Table of n, a(n) for n = 0..350

Row sums are A318566.
Column k = 1 is A000041 (for n > 0).
Column k = n is A007716.
Partitions of partitions of partitions are A007713.
Twice-factorizations are A050336.
The 2-dimensional version is A317533.
See A330472 for a variation.
Cf. A001055, A050336, A061260, A269134, A292504, A306186, A317791.

\\ See links in A339645 for combinatorial species functions.
ColGf(k, n)={my(A=symGroupSeries(n)); OgfSeries(sCartProd(sExp(A), sSubstOp(polcoef(sExp(A), k, x)*x^k + O(x*x^n), A) ))}
M(n, m=n)={Mat(vector(m+1, k, Col(ColGf(k-1, n), -(n+1))))}
{ my(A=M(10)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Jan 18 2023

Terms a(36) and beyond from Andrew Howroyd, Jan 18 2023

A335478 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (2,1,1).

Original entry on oeis.org

11, 19, 23, 27, 35, 39, 43, 45, 46, 47, 51, 55, 59, 67, 71, 74, 75, 77, 78, 79, 83, 87, 89, 91, 92, 93, 94, 95, 99, 103, 107, 109, 110, 111, 115, 119, 123, 131, 135, 138, 139, 141, 142, 143, 147, 149, 150, 151, 153, 154, 155, 156, 157, 158, 159, 163, 167, 171

Offset: 1

Gus Wiseman, Jun 18 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. 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.

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. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The sequence of terms together with the corresponding compositions begins:
  11: (2,1,1)
  19: (3,1,1)
  23: (2,1,1,1)
  27: (1,2,1,1)
  35: (4,1,1)
  39: (3,1,1,1)
  43: (2,2,1,1)
  45: (2,1,2,1)
  46: (2,1,1,2)
  47: (2,1,1,1,1)
  51: (1,3,1,1)
  55: (1,2,1,1,1)
  59: (1,1,2,1,1)
  67: (5,1,1)
  71: (4,1,1,1)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

The complement A335523 is the avoiding version.
The (1,1,2)-matching version is A335476.
Patterns matching this pattern are counted by A335509 (by length).
Permutations of prime indices matching this pattern are counted by A335516.
These compositions are counted by A335470 (by sum).
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335446, A335456, A335458, A335475.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],MatchQ[stc[#],{_,x_,_,y_,_,y_,_}/;x>y]&]

A335522 Numbers k such that the k-th composition in standard order (A066099) avoids the pattern (1,1,2).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 55, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81

Offset: 1

Gus Wiseman, Jun 18 2020

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.

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. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

Patterns avoiding this pattern are counted by A001710 (by length).
Permutations of prime indices avoiding this pattern are counted by A335449.
These compositions are counted by A335471 (by sum).
The complement A335476 is the matching version.
The (2,1,1)-avoiding version is A335523.
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335446, A335456, A335458.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],!MatchQ[stc[#],{_,x_,_,x_,_,y_,_}/;x

cp's OEIS Frontend

A316222 Number of positive subset-sum triangles whose composite is a positive subset-sum of an integer partition of n.

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A318567 Number of pairs (c, y) where c is an integer composition and y is an integer partition and y can be obtained from c by choosing a partition of each part, flattening, and sorting.

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A330472 Triangle read by rows where T(n,k) is the number of non-isomorphic k-element multisets of nonempty multisets of nonempty multisets (all finite).

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A335476 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,1,2).

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A335477 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (2,2,1).

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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.

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A318816 Regular tetrangle where T(n,k,i) is the number of non-isomorphic multiset partitions of length i of multiset partitions of length k of multisets of size n.

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A330473 Regular triangle where T(n,k) is the number of non-isomorphic multiset partitions of k-element multiset partitions of multisets of size n.

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PARI

Extensions

A335478 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (2,1,1).

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