cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A297568 Number of nonisomorphic proper colorings of partition star graph using four colors.

Original entry on oeis.org

4, 12, 24, 36, 40, 108, 108, 60, 216, 180, 324, 324, 84, 360, 540, 648, 972, 972, 972, 112, 540, 1080, 660, 1080, 2916, 1512, 1944, 2916, 2916, 2916, 144, 756, 1800, 1980, 1620, 5832, 4860, 4536, 3240, 8748, 8748, 5832, 8748, 8748, 8748, 180, 1008, 2700, 3960, 1980, 2268, 9720, 14580, 9072, 13608, 4860, 17496, 14580, 26244, 13284, 9720, 26244, 26244, 17496, 26244, 26244, 26244, 220, 1296, 3780, 6600, 5940, 3024, 14580
Offset: 0

Views

Author

Marko Riedel, Dec 31 2017

Keywords

Comments

A partition star graph consists of a multiset of paths with lengths given by the elements of the partition attached to a distinguished root node. The ordering of the partitions is by traversing antichains in Young's lattice bottom to top, left to right. Isomorphism refers to the automorphisms of the star graph corresponding to the partition.

Examples

			Rows are:
   4;
  12;
  24,  36;
  40, 108, 108;
  60, 216, 180, 324, 324;
  84, 360, 540, 648, 972, 972, 972;

Links

Crossrefs

Cf. A297567, A297569, A297570.
Row sums give 4*A144067.
Row lengths give A000041.

Programs

  • Maple
    b:= (n, i)-> `if`(n=0, [4], `if`(i<1, [], [seq(map(x-> x*
     binomial(3^i+j-1, j), b(n-i*j, i-1))[], j=0..n/i)])):
T:= n-> b(n$2)[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Jan 14 2018
  • Mathematica
    b[n_, i_] := If[n == 0, {4}, If[i<1, {}, Table[Map[Function[x, x*Binomial[ 3^i + j - 1, j]], b[n - i*j, i - 1]], {j, 0, n/i}]] // Flatten];
T[n_] := b[n, n];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 17 2018, after Alois P. Heinz *)

Formula

For a partition lambda we have the OCP: k Product_{p^v in lambda} C((k-1)^p+v-1, v). Here we have k=4.

A297569 Number of nonisomorphic proper colorings of partition star graph using five colors.

Original entry on oeis.org

5, 20, 50, 80, 100, 320, 320, 175, 800, 680, 1280, 1280, 280, 1600, 2720, 3200, 5120, 5120, 5120, 420, 2800, 6800, 4080, 6400, 20480, 10400, 12800, 20480, 20480, 20480, 600, 4480, 13600, 16320, 11200, 51200, 43520, 41600, 25600, 81920, 81920, 51200, 81920, 81920, 81920, 825, 6720, 23800, 40800, 19380, 17920, 102400, 174080, 104000, 166400, 44800, 204800, 174080, 327680, 164480, 102400, 327680, 327680, 204800, 327680, 327680
Offset: 0

Views

Author

Marko Riedel, Dec 31 2017

Keywords

Comments

A partition star graph consists of a multiset of paths with lengths given by the elements of the partition attached to a distinguished root node. The ordering of the partitions is by traversing antichains in Young's lattice bottom to top, left to right. Isomorphism refers to the automorphisms of the star graph corresponding to the partition.

Examples

			Rows are:
    5;
   20,
   50,   80;
  100,  320,  320;
  175,  800,  680, 1280, 1280;
  280, 1600, 2720, 3200, 5120, 5120, 5120;

Links

Crossrefs

Cf. A297567, A297568, A297570.
Row sums give 5*A144068.
Row lengths give A000041.

Programs

  • Maple
    b:= (n, i)-> `if`(n=0, [5], `if`(i<1, [], [seq(map(x-> x*
     binomial(4^i+j-1, j), b(n-i*j, i-1))[], j=0..n/i)])):
T:= n-> b(n$2)[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Jan 14 2018
  • Mathematica
    b[n_, i_] := If[n == 0, {5}, If[i<1, {}, Table[Map[Function[x, x*Binomial[ 4^i + j - 1, j]], b[n - i*j, i - 1]], {j, 0, n/i}]] // Flatten];
T[n_] := b[n, n];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 17 2018, after Alois P. Heinz *)

Formula

For a partition lambda we have the OCP: k Product_{p^v in lambda} C((k-1)^p+v-1, v). Here we have k=5.

A297570 Number of nonisomorphic proper colorings of partition star graph using six colors.

Original entry on oeis.org

6, 30, 90, 150, 210, 750, 750, 420, 2250, 1950, 3750, 3750, 756, 5250, 9750, 11250, 18750, 18750, 18750, 1260, 10500, 29250, 17550, 26250, 93750, 47250, 56250, 93750, 93750, 93750, 1980, 18900, 68250, 87750, 52500, 281250, 243750, 236250, 131250, 468750, 468750, 281250, 468750, 468750, 468750, 2970, 31500, 136500, 263250, 122850, 94500, 656250, 1218750, 708750, 1181250, 262500, 1406250, 1218750, 2343750, 1173750
Offset: 0

Views

Author

Marko Riedel, Dec 31 2017

Keywords

Comments

A partition star graph consists of a multiset of paths with lengths given by the elements of the partition attached to a distinguished root node. The ordering of the partitions is by traversing antichains in Young's lattice bottom to top, left to right. Isomorphism refers to the automorphisms of the star graph corresponding to the partition.

Examples

			Rows are:
    6;
   30;
   90,  150;
  210,  750,  750;
  420, 2250, 1950,  3750,  3750;
  756, 5250, 9750, 11250, 18750, 18750, 18750;

Links

Crossrefs

Cf. A297567, A297568, A297569.
Row sums give 6*A144069.
Row lengths give A000041.

Programs

  • Maple
    b:= (n, i)-> `if`(n=0, [6], `if`(i<1, [], [seq(map(x-> x*
     binomial(5^i+j-1, j), b(n-i*j, i-1))[], j=0..n/i)])):
T:= n-> b(n$2)[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Jan 14 2018
  • Mathematica
    b[n_, i_] := If[n == 0, {6}, If[i<1, {}, Table[Map[Function[x, x*Binomial[ 5^i + j - 1, j]], b[n - i*j, i - 1]], {j, 0, n/i}]] // Flatten];
T[n_] :=  b[n, n];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 17 2018, after Alois P. Heinz *)

Formula

For a partition lambda we have the OCP: k Product_{p^v in lambda} C((k-1)^p+v-1, v). Here we have k=6.

A298263 Number of nonisomorphic proper colorings of partition multicycle graph using three colors.

Original entry on oeis.org

1, 3, 6, 3, 10, 9, 2, 15, 18, 6, 6, 6, 21, 30, 18, 12, 6, 18, 6, 28, 45, 36, 10, 20, 18, 3, 36, 18, 18, 14, 36, 63, 60, 30, 30, 36, 12, 9, 60, 54, 12, 36, 18, 42, 18, 45, 84, 90, 60, 15, 42, 60, 36, 18, 9, 90, 108, 36, 36, 21, 60, 54, 12, 84, 42, 54, 36, 55, 108, 126, 100, 45, 56, 90, 72, 20, 30, 27, 4, 126, 180, 108, 72, 36, 63, 90, 108, 36, 36, 36, 140, 126, 28, 108, 54, 108, 58
Offset: 0

Views

Author

Marko Riedel, Jan 15 2018

Keywords

Comments

A partition multicycle graph consists of a multiset of cycles with lengths given by the elements of the partition where degenerate cycles on one node are taken to be singletons and on two nodes a pair of nodes connected by an edge. The ordering of the partitions is by traversing antichains in Young's lattice bottom to top, left to right. Isomorphism refers to the automorphisms of the multicycle graph corresponding to the partition, consisting of permutations of cycles of the same length combined with rotations of individual cycles (no dihedral symmetry).

Examples

			Rows are:
   1;
   3;
   6;
  10,  9,  2;
  15, 18,  6,  6, 6;
  21, 30, 18, 12, 6, 18, 6;

Links

Crossrefs

Cf. A297567, A297568, A297569, A297570, A298264, A298265, A298266.

Formula

For a partition lambda we have the OCP: Product_{p^v in lambda} C(Q_p(k)+v-1, v)
where Q_1(k) = k, Q_2(k) = k(k-1)/2 and for n>=3, Q_n(k) = (1/n) * Sum_{d|n} phi(n/d) P_d(k) with P_d(k) = (k-1)^d + (-1)^d (k-1). Here we have k=3.

A298264 Number of nonisomorphic proper colorings of partition multicycle graph using four colors.

Original entry on oeis.org

1, 4, 10, 6, 20, 24, 8, 35, 60, 21, 32, 24, 56, 120, 84, 80, 48, 96, 48, 84, 210, 210, 56, 160, 192, 36, 240, 144, 192, 130, 120, 336, 420, 224, 280, 480, 168, 144, 480, 576, 192, 480, 288, 520, 312, 165, 504, 735, 560, 126, 448, 960, 672, 360, 216, 840, 1440, 504, 768, 300, 960, 1152, 384, 1300, 780, 1248, 834, 220, 720, 1176, 1120, 504, 672, 1680, 1680, 448, 720, 864, 120, 1344, 2880, 2016, 1920, 1152, 1200, 1680, 2880, 1008, 1536, 1152, 2600, 3120, 1040, 3120, 1872, 3336, 2192
Offset: 0

Views

Author

Marko Riedel, Jan 15 2018

Keywords

Comments

A partition multicycle graph consists of a multiset of cycles with lengths given by the elements of the partition where degenerate cycles on one node are taken to be singletons and on two nodes a pair of nodes connected by an edge. The ordering of the partitions is by traversing antichains in Young's lattice bottom to top, left to right. Isomorphism refers to the automorphisms of the multicycle graph corresponding to the partition, consisting of permutations of cycles of the same length combined with rotations of individual cycles (no dihedral symmetry).

Examples

			Rows are:
   1;
   4;
  10,   6;
  20,  24,  8;
  35,  60, 21, 32, 24;
  56, 120, 84, 80, 48, 96, 48;

Links

Crossrefs

Cf. A297567, A297568, A297569, A297570, A298263, A298265, A298266.

Formula

For a partition lambda we have the OCP: Product_{p^v in lambda} C(Q_p(k)+v-1, v)
where Q_1(k) = k, Q_2(k) = k(k-1)/2 and for n>=3, Q_n(k) = (1/n) * Sum_{d|n} phi(n/d) P_d(k) with P_d(k) = (k-1)^d + (-1)^d (k-1). Here we have k=4.

A298265 Number of nonisomorphic proper colorings of partition multicycle graph using five colors.

Original entry on oeis.org

1, 5, 15, 10, 35, 50, 20, 70, 150, 55, 100, 70, 126, 350, 275, 300, 200, 350, 204, 210, 700, 825, 220, 700, 1000, 210, 1050, 700, 1020, 700, 330, 1260, 1925, 1100, 1400, 3000, 1100, 1050, 2450, 3500, 1400, 3060, 2040, 3500, 2340, 495, 2100, 3850, 3300, 715, 2520, 7000, 5500, 3150, 2100, 4900, 10500, 3850, 7000, 2485, 7140, 10200, 4080, 10500, 7000, 11700, 8230, 715, 3300, 6930, 7700, 3575, 4200, 14000, 16500, 4400, 7350, 10500, 1540, 8820, 24500, 19250, 21000, 14000, 12425, 14280, 30600, 11220, 20400, 14280, 24500, 35000, 14000, 35100, 23400, 41150, 29140
Offset: 0

Views

Author

Marko Riedel, Jan 15 2018

Keywords

Comments

A partition multicycle graph consists of a multiset of cycles with lengths given by the elements of the partition where degenerate cycles on one node are taken to be singletons and on two nodes a pair of nodes connected by an edge. The ordering of the partitions is by traversing antichains in Young's lattice bottom to top, left to right. Isomorphism refers to the automorphisms of the multicycle graph corresponding to the partition, consisting of permutations of cycles of the same length combined with rotations of individual cycles (no dihedral symmetry).

Examples

			Rows are:
    1;
    5;
   15,  10;
   35,  50,  20;
   70, 150,  55, 100,  70;
  126, 350, 275, 300, 200, 350, 204;

Links

Crossrefs

Cf. A297567, A297568, A297569, A297570, A298263, A298264, A298266.

Formula

For a partition lambda we have the OCP: Product_{p^v in lambda} C(Q_p(k)+v-1, v)
where Q_1(k) = k, Q_2(k) = k(k-1)/2 and for n>=3, Q_n(k) = (1/n) * Sum_{d|n} phi(n/d) P_d(k) with P_d(k) = (k-1)^d + (-1)^d (k-1). Here we have k=5.

A298266 Number of nonisomorphic proper colorings of partition multicycle graph using six colors.

Original entry on oeis.org

1, 6, 21, 15, 56, 90, 40, 126, 315, 120, 240, 165, 252, 840, 720, 840, 600, 990, 624, 462, 1890, 2520, 680, 2240, 3600, 820, 3465, 2475, 3744, 2635, 792, 3780, 6720, 4080, 5040, 12600, 4800, 4920, 9240, 14850, 6600, 13104, 9360, 15810, 11160, 1287, 6930, 15120, 14280, 3060, 10080, 33600, 28800, 17220, 12300, 20790, 51975, 19800, 39600, 13695, 34944, 56160, 24960, 55335, 39525, 66960, 48915, 2002, 11880, 30240, 38080, 18360, 18480, 75600, 100800, 27200, 45920, 73800, 11480, 41580, 138600, 118800, 138600, 99000, 82170, 78624, 196560, 74880, 149760, 102960, 147560, 237150, 105400, 234360, 167400, 293490, 217040
Offset: 0

Views

Author

Marko Riedel, Jan 15 2018

Keywords

Comments

A partition multicycle graph consists of a multiset of cycles with lengths given by the elements of the partition where degenerate cycles on one node are taken to be singletons and on two nodes a pair of nodes connected by an edge. The ordering of the partitions is by traversing antichains in Young's lattice bottom to top, left to right. Isomorphism refers to the automorphisms of the multicycle graph corresponding to the partition, consisting of permutations of cycles of the same length combined with rotations of individual cycles (no dihedral symmetry).

Examples

			Rows are:
    1;
    6;
   21,  15;
   56,  90,  40;
  126, 315, 120, 240, 165;
  252, 840, 720, 840, 600, 990, 624;

Links

Crossrefs

Cf. A297567, A297568, A297569, A297570, A298263, A298264, A298265.

Formula

For a partition lambda we have the OCP: Product_{p^v in lambda} C(Q_p(k)+v-1, v)
where Q_1(k) = k, Q_2(k) = k(k-1)/2 and for n>=3, Q_n(k) = (1/n) * Sum_{d|n} phi(n/d) P_d(k) with P_d(k) = (k-1)^d + (-1)^d (k-1). Here we have k=6.
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