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 1-3 of 3 results.

A300626 Number of inequivalent colorings of free pure symmetric multifunctions (with empty expressions allowed) with n positions.

Original entry on oeis.org

1, 1, 3, 11, 43, 187, 872, 4375, 23258, 130485, 767348, 4710715, 30070205, 198983975, 1361361925, 9607908808, 69812787049, 521377973359, 3996036977270, 31389624598631, 252408597286705, 2075472033455894, 17434190966525003, 149476993511444023, 1307022313790487959
Offset: 0

Views

Author

Gus Wiseman, Aug 17 2018

Keywords

Comments

A free pure symmetric multifunction (with empty expressions allowed) f in EOME is either (case 1) a positive integer, or (case 2) a possibly empty expression of the form h[g_1, ..., g_k] where k >= 0, h is in EOME, each of the g_i for i = 1, ..., k is in EOME, and for i < j we have g_i <= g_j under a canonical total ordering of EOME, such as the Mathematica ordering of expressions.
Also the number of inequivalent colorings of orderless Mathematica expressions with n positions.

Examples

			Inequivalent representatives of the a(3) = 11 colorings:
  1[1,1]  1[2,2]  1[1,2]  1[2,3]
  1[1[]]  1[2[]]
  1[][1]  1[][2]
  1[1][]  1[2][]
  1[][][]

Crossrefs

Row sums of A304485.
Cf. A000612, A007716, A052893, A053492, A277996, A279944, A280000, A317652, A317655, A317656, A317676.

Programs

  • PARI
    \\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(p=O(x)); for(n=1, n, p = x*sv(1) + x*p*sExp(p)); p}
InequivalentColoringsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 30 2020

Extensions

Terms a(8) and beyond from Andrew Howroyd, Dec 30 2020

A304485 Regular triangle where T(n,k) is the number of inequivalent colorings of free pure symmetric multifunctions (with empty expressions allowed) with n positions and k leaves.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 6, 4, 0, 1, 12, 23, 7, 0, 1, 20, 81, 73, 12, 0, 1, 30, 209, 407, 206, 19, 0, 1, 42, 451, 1566, 1751, 534, 30, 0, 1, 56, 858, 4711, 9593, 6695, 1299, 45, 0, 1, 72, 1494, 11951, 39255, 51111, 23530, 3004, 67, 0, 1, 90, 2430, 26752, 130220, 278570, 245319, 77205, 6664, 97, 0
Offset: 1

Views

Author

Gus Wiseman, Aug 17 2018

Keywords

Comments

A free pure symmetric multifunction (with empty expressions allowed) f in EOME is either (case 1) a positive integer, or (case 2) a possibly empty expression of the form h[g_1, ..., g_k] where k >= 0, h is in EOME, each of the g_i for i = 1, ..., k is in EOME, and for i < j we have g_i <= g_j under a canonical total ordering of EOME, such as the Mathematica ordering of expressions.
T(n,k) is also the number of inequivalent colorings of orderless Mathematica expressions with n positions and k leaves.

Examples

			Inequivalent representatives of the T(5,3) = 23 Mathematica expressions:
  1[][1,1]  1[1,1][]  1[1][1]  1[1[1]]  1[1,1[]]
  1[][1,2]  1[1,2][]  1[1][2]  1[1[2]]  1[1,2[]]
  1[][2,2]  1[2,2][]  1[2][1]  1[2[1]]  1[2,1[]]
  1[][2,3]  1[2,3][]  1[2][2]  1[2[2]]  1[2,2[]]
                      1[2][3]  1[2[3]]  1[2,3[]]
Triangle begins:
    1
    1    0
    1    2    0
    1    6    4    0
    1   12   23    7    0
    1   20   81   73   12    0
    1   30  209  407  206   19    0
    1   42  451 1566 1751  534   30    0

Links

Crossrefs

Row sums are A300626.
Cf. A000612, A007716, A052893, A053492, A277996, A280000, A317676.

Programs

  • PARI
    \\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(p=O(x)); for(n=1, n, p = x*sv(1) + x*p*sExp(p)); p}
T(n)={my(v=Vec(InequivalentColoringsSeq(sFuncSubst(cycleIndexSeries(n), i->sv(i)*y^i)))); vector(n, n, Vecrev(v[n]/y, n))}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 01 2021

Extensions

Terms a(37) and beyond from Andrew Howroyd, Jan 01 2021

A317659 Regular triangle where T(n,k) is the number of distinct free pure symmetric multifunctions (with empty expressions allowed) with one atom, n positions, and k leaves.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 5, 1, 0, 1, 10, 17, 7, 1, 0, 1, 15, 43, 33, 9, 1, 0, 1, 21, 92, 118, 55, 11, 1, 0, 1, 28, 174, 341, 252, 82, 13, 1, 0, 1, 36, 302, 845, 935, 463, 115, 15, 1, 0, 1, 45, 490, 1864, 2921, 2103, 769, 153, 17, 1, 0, 1, 55, 755
Offset: 1

Views

Author

Gus Wiseman, Aug 03 2018

Keywords

Examples

			The T(5,3) = 5 expressions are o[o[o]], o[o,o[]], o[][o,o], o[o][o], o[o,o][].
Triangle begins:
    1
    1    0
    1    1    0
    1    3    1    0
    1    6    5    1    0
    1   10   17    7    1    0
    1   15   43   33    9    1    0
    1   21   92  118   55   11    1    0
    1   28  174  341  252   82   13    1    0
    1   36  302  845  935  463  115   15    1    0
    1   45  490 1864 2921 2103  769  153   17    1    0
    1   55  755 3755 7981 8012 4145 1187  197   19    1    0

Links

Crossrefs

Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000.
Cf. A317652, A317653, A317654, A317655, A317656, A317676.

Programs

  • Mathematica
    maxUsing[n_]:=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-1}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{maxUsing[h],Union[Sort/@Tuples[maxUsing/@p]]}],{p,IntegerPartitions[g]}]]];
Table[Length[Select[maxUsing[n],Length[Position[#,"o"]]==k&]],{n,12},{k,n}]
Showing 1-3 of 3 results.

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