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.

Previous Showing 11-15 of 15 results.

A320664 Number of non-isomorphic multiset partitions of weight n with all parts of odd size.

Original entry on oeis.org

1, 1, 2, 6, 12, 30, 82, 198, 533, 1459, 4039, 11634, 34095, 102520, 316456, 995709, 3215552, 10591412, 35633438, 122499429, 428988392, 1532929060, 5579867442, 20677066725, 78027003260, 299413756170, 1168536196157, 4635420192861, 18678567555721, 76451691937279, 317625507668759
Offset: 0

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Author

Gus Wiseman, Oct 18 2018

Keywords

Comments

Also the number of non-isomorphic multiset partitions of weight n in which each vertex appears an odd number of times.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 12 multiset partitions with all parts of odd size:
  {{1}}  {{1},{1}}  {{1,1,1}}      {{1},{1,1,1}}
         {{1},{2}}  {{1,2,2}}      {{1},{1,2,2}}
                    {{1,2,3}}      {{1},{2,2,2}}
                    {{1},{1},{1}}  {{1},{2,3,3}}
                    {{1},{2},{2}}  {{1},{2,3,4}}
                    {{1},{2},{3}}  {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{2},{2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}

Links

Crossrefs

Cf. A001055, A001222, A007716, A298118, A300300, A300301, A318871, A320663, A320665.

Programs

  • PARI
    \\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n)); NumUnlabeledObjsSeq(sCartProd(sExp(A), sExp((A-subst(A,x,-x))/2)))} \\ Andrew Howroyd, Jan 17 2023
  • PARI
    permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
J(q, t, k, y)={1/prod(j=1, #q, my(s=q[j], g=gcd(s,t)); (1 + O(x*x^k) - y^(s/g)*x^(s*t/g))^g)}
K(q, t, k) = Vec(J(q,t,k,1)-J(q,t,k,-1), -k)/2
a(n)={my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(sum(t=1, n, K(q, t, n)/t))), n)); s/n!} \\ Andrew Howroyd, Jan 17 2023

Extensions

Terms a(11) and beyond from Andrew Howroyd, Jan 16 2023

A330057 Number of set-systems covering n vertices with no singletons or endpoints.

Original entry on oeis.org

1, 0, 0, 5, 1703, 66954642, 144115175199102143, 1329227995784915808340204290157341181, 226156424291633194186662080095093568664788471116325389572604136316742486364
Offset: 0

Views

Author

Gus Wiseman, Nov 30 2019

Keywords

Comments

A set-system is a finite set of finite nonempty set of positive integers. A singleton is an edge of size 1. An endpoint is a vertex appearing only once (degree 1).

Examples

			The a(3) = 5 set-systems:
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{1,2,3}}
  {{1,2},{2,3},{1,2,3}}
  {{1,3},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}

Links

Crossrefs

The version for non-isomorphic set-systems is A330055 (by weight).
The non-covering version is A330056.
Set-systems with no singletons are A016031.
Set-systems with no endpoints are A330059.
Non-isomorphic set-systems with no singletons are A306005 (by weight).
Non-isomorphic set-systems with no endpoints are A330054 (by weight).
Non-isomorphic set-systems counted by vertices are A000612.
Non-isomorphic set-systems counted by weight are A283877.
Cf. A007716, A055621, A302545, A317533, A317794, A319559, A320665, A321405, A330052, A330058.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Subsets[Range[n],{2,n}]],Union@@#==Range[n]&&Min@@Length/@Split[Sort[Join@@#]]>1&]],{n,0,4}]
  • PARI
    \\ here b(n) is A330056(n).
AS2(n, k) = {sum(i=0, min(n, k), (-1)^i * binomial(n, i) * stirling(n-i, k-i, 2) )}
b(n) = {sum(k=0, n, (-1)^k*binomial(n,k)*2^(2^(n-k)-(n-k)-1) * sum(j=0, k\2, sum(i=0, k-2*j, binomial(k,i) * AS2(k-i, j) * (2^(n-k)-1)^i * 2^(j*(n-k)) )))}
a(n) = {sum(k=0, n, (-1)^k*binomial(n,k)*b(n-k))} \\ Andrew Howroyd, Jan 16 2023

Formula

Binomial transform is A330056.

Extensions

Terms a(5) and beyond from Andrew Howroyd, Jan 16 2023

A369287 Triangle read by rows: T(n,k) is the number of non-isomorphic multiset partitions of weight n with k parts and no singletons or vertices that appear only once, 0 <= k <= floor(n/2).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 2, 3, 0, 2, 4, 0, 4, 15, 8, 0, 4, 24, 19, 0, 7, 60, 79, 23, 0, 8, 101, 213, 84, 0, 12, 210, 615, 424, 66, 0, 14, 357, 1523, 1533, 363, 0, 21, 679, 3851, 5580, 2217, 212, 0, 24, 1142, 8963, 17836, 10379, 1575, 0, 34, 2049, 20840, 55730, 45866, 11616, 686
Offset: 0

Views

Author

Andrew Howroyd, Jan 28 2024

Keywords

Comments

T(n,k) is the number of nonnegative integer matrices with sum of values n, k rows and every row and column sum at least two up to permutation of rows and columns.

Examples

			Triangle begins:
  1;
  0;
  0,  1;
  0,  1;
  0,  2,    3;
  0,  2,    4;
  0,  4,   15,    8;
  0,  4,   24,   19;
  0,  7,   60,   79,    23;
  0,  8,  101,  213,    84;
  0, 12,  210,  615,   424,    66;
  0, 14,  357, 1523,  1533,   363;
  0, 21,  679, 3851,  5580,  2217,  212;
  0, 24, 1142, 8963, 17836, 10379, 1575;
  ...
The T(5,1) = 2 multiset partitions are:
   {{1,1,1,1,1}},
   {{1,1,1,2,2}}.
The corresponding T(5,1) = 2 matrices are:
   [5]  [3 2].
The T(5,2) = 4 matrices are:
   [3]  [3 0]  [2 1]  [2 1]
   [2]  [0 2]  [1 1]  [0 2],

Links

Crossrefs

Row sums are A320665.
Columns k=0..1 are A000007, A002865(n>0).
Cf. A050535, A369286.

Programs

  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
H(q, t, k)={my(c=sum(j=1, #q, if(t%q[j]==0, q[j]))); (1 - x)*x*Ser(K(q,t,k)) - c*x}
G(n,y=1)={my(s=0); forpart(q=n, s+=permcount(q)*exp(sum(t=1, n, subst(H(q, t, n\t)*y^t/t, x, x^t) ))); s/n!}
T(n)={my(v=Vec(G(n,'y))); vector(#v, i, Vecrev(v[i], (i+1)\2))}
{ my(A=T(15)); for(i=1, #A, print(A[i])) }

Formula

T(2*n, n) = A050535(n).

A330124 Number of unlabeled set-systems with n vertices and no endpoints.

Original entry on oeis.org

1, 1, 2, 22, 1776
Offset: 0

Views

Author

Gus Wiseman, Dec 05 2019

Keywords

Comments

A set-system is a finite set of finite nonempty sets. An endpoint is a vertex appearing only once (degree 1).

Examples

			Non-isomorphic representatives of the a(3) = 22 set-systems:
  0
  {1}{2}{12}
  {12}{13}{23}
  {1}{23}{123}
  {12}{13}{123}
  {1}{2}{13}{23}
  {1}{2}{3}{123}
  {1}{12}{13}{23}
  {1}{2}{13}{123}
  {1}{12}{13}{123}
  {1}{12}{23}{123}
  {12}{13}{23}{123}
  {1}{2}{3}{12}{13}
  {1}{2}{12}{13}{23}
  {1}{2}{3}{12}{123}
  {1}{2}{12}{13}{123}
  {1}{2}{13}{23}{123}
  {1}{12}{13}{23}{123}
  {1}{2}{3}{12}{13}{23}
  {1}{2}{3}{12}{13}{123}
  {1}{2}{12}{13}{23}{123}
  {1}{2}{3}{12}{13}{23}{123}

Crossrefs

Partial sums of the covering case A330196.
The labeled version is A330059.
The "multi" version is A302545.
Unlabeled set-systems with no endpoints counted by weight are A330054.
Unlabeled set-systems with no singletons are A317794.
Unlabeled set-systems counted by vertices are A000612.
Unlabeled set-systems counted by weight are A283877.
The case with no singletons is A320665.
Cf. A007716, A016031, A306005, A317795, A321405, A330052, A330055.

A330196 Number of unlabeled set-systems covering n vertices with no endpoints.

Original entry on oeis.org

1, 0, 1, 20, 1754
Offset: 0

Views

Author

Gus Wiseman, Dec 05 2019

Keywords

Comments

A set-system is a finite set of finite nonempty sets. An endpoint is a vertex appearing only once (degree 1).

Examples

			Non-isomorphic representatives of the a(3) = 20 set-systems:
  {12}{13}{23}
  {1}{23}{123}
  {12}{13}{123}
  {1}{2}{13}{23}
  {1}{2}{3}{123}
  {1}{12}{13}{23}
  {1}{2}{13}{123}
  {1}{12}{13}{123}
  {1}{12}{23}{123}
  {12}{13}{23}{123}
  {1}{2}{3}{12}{13}
  {1}{2}{12}{13}{23}
  {1}{2}{3}{12}{123}
  {1}{2}{12}{13}{123}
  {1}{2}{13}{23}{123}
  {1}{12}{13}{23}{123}
  {1}{2}{3}{12}{13}{23}
  {1}{2}{3}{12}{13}{123}
  {1}{2}{12}{13}{23}{123}
  {1}{2}{3}{12}{13}{23}{123}

Crossrefs

First differences of the non-covering version A330124.
The "multi" version is A302545.
Unlabeled set-systems with no endpoints counted by vertices are A317794.
Unlabeled set-systems with no endpoints counted by weight are A330054.
Unlabeled set-systems counted by vertices are A000612.
Unlabeled set-systems counted by weight are A283877.
Cf. A007716, A016031, A306005, A317795, A320665, A330052, A330053, A330055.
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