A117662 -id:A117662 - cp's OEIS frontend

A289223 Number of ways to select 2 disjoint point triples from an n X n X n triangular point grid, each point triple forming an 2 X 2 X 2 triangle.

0, 0, 12, 66, 204, 480, 960, 1722, 2856, 4464, 6660, 9570, 13332, 18096, 24024, 31290, 40080, 50592, 63036, 77634, 94620, 114240, 136752, 162426, 191544, 224400, 261300, 302562, 348516, 399504, 455880, 518010, 586272, 661056, 742764, 831810, 928620, 1033632, 1147296

Offset: 2

Heinrich Ludwig, Jun 28 2017

Rotations and reflections of a selection are regarded as different. For the number of congruence classes see A117662(n-1).

			There are 12 ways to choose two 2 X 2 X 2 triangles (xxx) from a 4 X 4 X 4 point grid, for example:
      x           x          x
     x x         x x        x x
    . x x       x . .      . x .
   . . x .     x x . .    . x x .
The other nine selections are reflections or rotations of these three.

Heinrich Ludwig, Table of n, a(n) for n = 2..100
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A117662, A289222, A289224, A289225, A289226, A289227, A289228.

Vec(6*x^4*(2 - x)*(1 + x) / (1 - x)^5 + O(x^60)) \\ Colin Barker, Jun 28 2017

a(n) = (n^4 -4*n^3 -7*n^2 +46*n -48)/2 for n>=2.
From Colin Barker, Jun 28 2017: (Start)
G.f.: 6*x^4*(2 - x)*(1 + x) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>6.
(End)

A289229 Triangle read by rows: T(n, k) is the number of nonequivalent ways to select k disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 3, 3, 2, 1, 5, 14, 19, 4, 0, 1, 7, 40, 127, 159, 77, 17, 0, 1, 9, 90, 536, 1644, 2569, 1876, 500, 42, 1, 1, 12, 175, 1688, 9548, 31951, 62171, 67765, 39459, 11579, 1547, 47, 0, 1, 15, 308, 4357, 38872, 223346, 832628, 2005948, 3072004, 2897626

Offset: 1

Heinrich Ludwig, Jul 04 2017

The row index starts from 1. The column index k runs from 0 to floor(n*(n+1)/6), which is a trivial upper bound for the maximal number of 2 X 2 X 2 triangles that can be selected from an n X n X n triangular grid.

Rotations and reflections of a selection are not counted. If they are to be counted, see A289222.

			The triangle begins:
  1;
  1,  1;
  1,  2,   0;
  1,  3,   3,    3;
  1,  5,  14,   19,    4,     0;
  1,  7,  40,  127,  159,    77,    17,     0;
  1,  9,  90,  536, 1644,  2569,  1876,   500,    42,     1;
  1, 12, 175, 1688, 9548, 31951, 62171, 67765, 39459, 11579, 1547, 47, 0;

Heinrich Ludwig, Table of n, a(n) for n = 1..116, first 11 (and a half) rows of the triangular array

Cf. A289222, A289233.

Columns 2 to 6: A001840, A117662, A289230, A289231, A289232.

A326331 Number of simple graphs covering the vertices {1..n} whose nesting edges are connected.

Original entry on oeis.org

1, 0, 1, 0, 1, 14, 539

Offset: 0

Gus Wiseman, Jun 27 2019

Covering means there are no isolated vertices. Two edges {a,b}, {c,d} are nesting if a < c < d < b or c < a < b < d. A graph has its nesting edges connected if the graph whose vertices are the edges and whose edges are nesting pairs of edges is connected.

Gus Wiseman, The a(5) = 14 simple nesting-connected covering graphs.

The non-covering case is the binomial transform A326330.
Covering graphs whose crossing edges are connected are A324327.
Cf. A006125, A007297, A054726, A099947, A117662, A136653, A324328.
Cf. A326210, A326293, A326335, A326336, A326337, A326338, A326339.

nesXQ[stn_]:=MatchQ[stn,{_,{x_,y_},_,{z_,t_},_}/;x0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[nestcmpts[#]]<=1&]],{n,0,5}]

A326339 Number of connected simple graphs with vertices {1..n} and no crossing or nesting edges.

Original entry on oeis.org

1, 0, 1, 4, 12, 36, 108, 324

Offset: 0

Gus Wiseman, Jun 29 2019

Two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b, and nesting if a < c < d < b or c < a < b < d.

Appears to be essentially the same as A003946.

			The a(2) = 1 through a(4) = 36 edge-sets:
  {12}  {12,13}     {12,13,14}
        {12,23}     {12,13,34}
        {13,23}     {12,14,34}
        {12,13,23}  {12,23,24}
                    {12,23,34}
                    {12,24,34}
                    {13,23,34}
                    {14,24,34}
                    {12,13,14,34}
                    {12,13,23,34}
                    {12,14,24,34}
                    {12,23,24,34}

Gus Wiseman, The a(5) = 36 connected graphs with no crossing or nesting edges.
Gus Wiseman, The a(6) = 108 connected graphs with no crossing or nesting edges.

Covering graphs with no crossing or nesting edges are A326329.
Connected simple graphs are A001349.
The case with only crossing edges forbidden is A007297.
Graphs without crossing or nesting edges are A326244.
Cf. A006125, A054726, A117662, A136653.
Cf. A324169, A326210, A326293, A326340.

csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[csm[#]]<=1&&!MatchQ[#,{_,{x_,y_},_,{z_,t_},_}/;x

A326340 Number of maximal simple graphs with vertices {1..n} and no crossing or nesting edges.

Original entry on oeis.org

1, 1, 1, 1, 4, 9, 19, 42

Offset: 0

Gus Wiseman, Jun 29 2019

Two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b, and nesting if a < c < d < b or c < a < b < d.

Gus Wiseman, The a(6) = 19 maximal non-crossing non-nesting graphs.

Covering graphs with no crossing or nesting edges are A326329.
The case with only crossing edges forbidden is A000108 shifted right twice.
Simple graphs without crossing or nesting edges are A326244.
Connected graphs with no crossing or nesting edges are A326339.
Cf. A006125, A007297, A054726, A117662, A136653.
Cf. A324169, A324327, A324328, A326279, A326293.

fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Subsets[Range[n],{2}]],!MatchQ[#,{_,{x_,y_},_,{z_,t_},_}/;x

A289230 Number of nonequivalent ways to select 3 disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.

Original entry on oeis.org

0, 2, 19, 127, 536, 1688, 4357, 9789, 19844, 37172, 65397, 109335, 175214, 270934, 406329, 593463, 846934, 1184212, 1625979, 2196509, 2924050, 3841240, 4985531, 6399647, 8132044, 10237410, 12777167, 15820007, 19442436, 23729352, 28774625, 34681717, 41564304, 49546932

Offset: 3

Heinrich Ludwig, Jun 30 2017

Rotations and reflections of a selection are not counted. If they are to be counted see A289224.

			There are two nonequivalent ways to choose three 2 X 2 X 2 triangles (aaa, bbb, ccc) from a 4 X 4 X 4 point grid:
      a           a
     a a         a a
    b c c       b . c
   b b c .     b b c c
Note: aaa, bbb, ccc are not distinguishable, they are denoted differently for a better perception of the 2 X 2 X 2 triangles only.

Heinrich Ludwig, Table of n, a(n) for n = 3..100
Index entries for linear recurrences with constant coefficients, signature (5,-9,6,0,0,0,-6,9,-5,1).

Cf. A289224, A289229, A117662, A289231, A289232.

concat(0, Vec(x^4*(2 + 9*x + 50*x^2 + 60*x^3 + 37*x^4 - 21*x^5 - 20*x^6 - 4*x^7 + 7*x^8) / ((1 - x)^7*(1 + x)*(1 + x + x^2)) + O(x^40))) \\ Colin Barker, Jun 30 2017

a(n) = (n^6 -6*n^5 -24*n^4 +220*n^3 -153*n^2 -1488*n +2592)/36 + IF(MOD(n, 2) = 1, -1)/2 + IF(MOD(n, 3) = 1, -2)/9.

G.f.: x^4*(2 + 9*x + 50*x^2 + 60*x^3 + 37*x^4 - 21*x^5 - 20*x^6 - 4*x^7 + 7*x^8) / ((1 - x)^7*(1 + x)*(1 + x + x^2)). - Colin Barker, Jun 30 2017

A289231 Number of nonequivalent ways to select 4 disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.

Original entry on oeis.org

0, 4, 159, 1644, 9548, 38872, 125367, 342831, 829052, 1822785, 3714519, 7113539, 12935256, 22511616, 37728563, 61194888, 96446684, 148191316, 222597315, 327633979, 473466444, 672912717, 941968139, 1300402591, 1772439504, 2387521212, 3181168199, 4195941108, 5482512012

Offset: 4

Heinrich Ludwig, Jun 30 2017

Rotations and reflections of a selection are not counted. If they are to be counted see A289225.

			There are four nonequivalent ways to choose four 2 X 2 X 2 triangles (aaa, ..., ddd) from a 5 X 5 X 5 point grid:
      a           a           a           .
     a a         a a         a a         a a
    b c c       . d .       . . .       . a .
   b b c d     b d d c     b c c d     b c c d
  . . . d d   b b . c c   b b c d d   b b c d d
Note: aaa, ..., ddd are not distinguishable, they are denoted differently for a better perception of the 2 X 2 X 2 triangles only.

Heinrich Ludwig, Table of n, a(n) for n = 4..100
Index entries for linear recurrences with constant coefficients, signature (4,-4,-1,-2,8,5,-14,-1,1,14,-5,-8,2,1,4,-4,1).

Cf. A289229, A289225, A117662, A289230, A289232.

concat(0, Vec(x^5*(4 + 143*x + 1024*x^2 + 3612*x^3 + 7423*x^4 + 10001*x^5 + 8395*x^6 + 3273*x^7 - 1362*x^8 - 2393*x^9 - 878*x^10 + 488*x^11 + 539*x^12 + 101*x^13 - 89*x^14 - 41*x^15) / ((1 - x)^9*(1 + x)^2*(1 + x + x^2)^3) + O(x^40))) \\ Colin Barker, Jun 30 2017

a(n) = (n^8 -8*n^7 -50*n^6 +556*n^5 +261*n^4 -12724*n^3 +19088*n^2 +86016*n -201024)/144 + IF(MOD(n, 2) = 1, -2*n +5)/4 + IF(MOD(n, 3) = 1, -n^2 +2*n +12)/9.

G.f.: x^5*(4 + 143*x + 1024*x^2 + 3612*x^3 + 7423*x^4 + 10001*x^5 + 8395*x^6 + 3273*x^7 - 1362*x^8 - 2393*x^9 - 878*x^10 + 488*x^11 + 539*x^12 + 101*x^13 - 89*x^14 - 41*x^15) / ((1 - x)^9*(1 + x)^2*(1 + x + x^2)^3). - Colin Barker, Jun 30 2017

A289232 Number of nonequivalent ways to select 5 disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.

Original entry on oeis.org

0, 77, 2569, 31951, 223346, 1089665, 4161705, 13314461, 37246668, 93781829, 216901737, 467727523, 951014654, 1839155785, 3406165049, 6074688977, 10479716856, 17553399741, 28636182537, 45620375447, 71133273514, 108768061009, 163371926729, 241402171109, 351362501892

Offset: 5

Heinrich Ludwig, Jul 01 2017

Rotations and reflections of a selection are not counted. If they are to be counted see A289226.

			There are 77 nonequivalent ways to choose five 2 X 2 X 2 triangles (aaa, ..., eee) from a 6 X 6 X 6 point grid, for example:
        .               a
       . .             a a
      . . .           . d .
     a a b b         b d d c
    c a d b e       b b e c c
   c c d d e e     . . e e . .
Note: aaa, ..., eee are not distinguishable, they are denoted differently for a better perception of the 2 X 2 X 2 triangles only.

Heinrich Ludwig, Table of n, a(n) for n = 5..100
Index entries for linear recurrences with constant coefficients, signature (8,-25,32,11,-88,99,0,-99,88,-11,-32,25,-8,1).

Cf. A289229, A117662, A289230, A289231, A289226.

concat(0, Vec(x^6*(77 + 1953*x + 13324*x^2 + 29499*x^3 + 18617*x^4 - 15880*x^5 - 17638*x^6 + 4876*x^7 + 8057*x^8 - 881*x^9 - 1966*x^10 + 81*x^11 + 201*x^12) / ((1 - x)^11*(1 + x)^3) + O(x^40))) \\ Colin Barker, Jul 01 2017

a(n) = (n^10 -10*n^9 -85*n^8 +1160*n^7 +1345*n^6 -49084*n^5 +61035*n^4 +897210*n^3 -2205196*n^2 -5725656*n +18174960)/720 + IF(MOD(n, 2) = 1, -2*n^2 +13*n -11)/4.

G.f.: x^6*(77 + 1953*x + 13324*x^2 + 29499*x^3 + 18617*x^4 - 15880*x^5 - 17638*x^6 + 4876*x^7 + 8057*x^8 - 881*x^9 - 1966*x^10 + 81*x^11 + 201*x^12) / ((1 - x)^11*(1 + x)^3). - Colin Barker, Jul 01 2017

A208518 Triangle of coefficients of polynomials u(n,x) jointly generated with A208519; see the Formula section.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 7, 3, 1, 10, 16, 14, 5, 1, 15, 30, 40, 28, 8, 1, 21, 50, 90, 93, 53, 13, 1, 28, 77, 175, 238, 203, 99, 21, 1, 36, 112, 308, 518, 588, 428, 181, 34, 1, 45, 156, 504, 1008, 1428, 1380, 873, 327, 55, 1, 55, 210, 780, 1806, 3066, 3690, 3105

Offset: 1

Clark Kimberling, Feb 28 2012

coefficient of x^(n-1): = Fibonacci(n) = A000045(n)
col 1:  A000012
col 2:  A000217 (triangular numbers)
col 3:  A005581
col 4:  A117662
alternating row sums: signed version of (-1+Fibonacci(n))

			First five rows:
1
1...1
1...3....2
1...6....7....3
1...10...16...14...5
First five polynomials u(n,x):
1
1 + x
1 + 3x + 2x^2
1 + 6x + 7x^2 + 3x^3
1 + 10x + 16x^2 + 14x^3 + 5x^4

Cf. A208519.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A208518 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A208519 *)

u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A326260 MM-numbers of capturing, non-nesting multiset partitions (with empty parts allowed).

Original entry on oeis.org

2599, 4163, 5198, 6463, 6893, 7291, 7797, 8326, 8507, 9131, 9959, 10396, 10649, 11041, 11639, 12489, 12811, 12926, 12995, 13786, 14237, 14582, 14899, 15157, 15594, 16123, 16403, 16652, 17014, 17063, 17089, 17141, 18101, 18193, 18262, 18643, 18659, 19337, 19389

Offset: 1

Gus Wiseman, Jun 22 2019

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 multiset multisystem with MM-number n is obtained by taking the multiset of prime indices of each prime index of n.

A set partition is capturing if it has two blocks of the form {...x...y...} and {...z...t...} where x < z and y > t or x > z and y < t. It is nesting if it has two blocks of the form {...x,y...} and {...z,t...} where x < z and y > t or x > z and y < t. Capturing is a weaker condition than nesting, so for example {{1,3,5},{2,4}} is capturing but not nesting.

			The sequence of terms together with their multiset multisystems begins:
   2599: {{2,2},{1,2,3}}
   4163: {{2,2},{1,2,4}}
   5198: {{},{2,2},{1,2,3}}
   6463: {{2,2},{1,1,2,3}}
   6893: {{1,2,2},{1,2,3}}
   7291: {{2,2},{1,2,5}}
   7797: {{1},{2,2},{1,2,3}}
   8326: {{},{2,2},{1,2,4}}
   8507: {{2,3},{1,2,4}}
   9131: {{2,2},{1,2,6}}
   9959: {{2,2},{1,1,2,4}}
  10396: {{},{},{2,2},{1,2,3}}
  10649: {{2,2},{1,2,2,3}}
  11041: {{1,2,2},{1,2,4}}
  11639: {{2,2,2},{1,2,3}}
  12489: {{1},{2,2},{1,2,4}}
  12811: {{2,2},{1,2,7}}
  12926: {{},{2,2},{1,1,2,3}}
  12995: {{2},{2,2},{1,2,3}}
  13786: {{},{1,2,2},{1,2,3}}

Non-nesting set partitions are A000108.
Capturing set partitions are A326243.
Capturing, non-nesting set partitions are A326249.
MM-numbers of nesting multiset partitions are A326256.
MM-numbers of capturing multiset partitions are A326255.
Cf. A001055, A034827, A056239, A058681, A112798, A117662, A302242, A324170.
Cf. A326212, A326246, A054391, A326257, A326258, A326259.

capXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x_,{_,x_,y_,_},_,{_,z_,t_,_},_}/;xTable[PrimePi[p],{k}]]]];
Select[Range[10000],!nesXQ[primeMS/@primeMS[#]]&&capXQ[primeMS/@primeMS[#]]&]

cp's OEIS Frontend

A289223 Number of ways to select 2 disjoint point triples from an n X n X n triangular point grid, each point triple forming an 2 X 2 X 2 triangle.

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A289229 Triangle read by rows: T(n, k) is the number of nonequivalent ways to select k disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.

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A326331 Number of simple graphs covering the vertices {1..n} whose nesting edges are connected.

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A326339 Number of connected simple graphs with vertices {1..n} and no crossing or nesting edges.

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A326340 Number of maximal simple graphs with vertices {1..n} and no crossing or nesting edges.

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A289230 Number of nonequivalent ways to select 3 disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.

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A289231 Number of nonequivalent ways to select 4 disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.

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A289232 Number of nonequivalent ways to select 5 disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.

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