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 1-6 of 6 results.

A362827 Array read by antidiagonals: T(n,k) is the number of k-tuples of permutations of [n] that pairwise commute.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 6, 1, 1, 1, 8, 18, 24, 1, 1, 1, 16, 48, 120, 120, 1, 1, 1, 32, 126, 504, 840, 720, 1, 1, 1, 64, 336, 2016, 4680, 7920, 5040, 1, 1, 1, 128, 918, 7944, 24720, 66240, 75600, 40320, 1, 1, 1, 256, 2568, 31200, 130440, 516240, 856800, 887040, 362880, 1
Offset: 0

Views

Author

Andrew Howroyd, May 08 2023

Keywords

Comments

Two permutations x,y on [n] commute if x*y = y*x.

Examples

			Array begins:
========================================================
n/k| 0    1     2      3       4        5          6 ...
---+----------------------------------------------------
0  | 1    1     1      1       1        1          1 ...
1  | 1    1     1      1       1        1          1 ...
2  | 1    2     4      8      16       32         64 ...
3  | 1    6    18     48     126      336        918 ...
4  | 1   24   120    504    2016     7944      31200 ...
5  | 1  120   840   4680   24720   130440     699840 ...
6  | 1  720  7920  66240  516240  3968640   30672720 ...
7  | 1 5040 75600 856800 9122400 97030080 1050336000 ...
  ...

Links

Crossrefs

Columns k=0..3 are A000012, A000142, A053529, A072169.
Main diagonal is A362828.
Cf. A362824, A362826.

Programs

  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
M(n,m=n)={my(v=vector(m+1), u=vector(n,n,n==1), f=vector(n,n,n!)); v[1]=vectorv(n+1,i,1); for(j=1, #v-1, my(t=EulerT(u)); v[j+1]=vectorv(n+1,i,i--;if(i,f[i]*t[i],1)); u=dirmul(u, vector(n, n, n^(j-1)))); Mat(v)}
{ my(A=M(7)); for(n=1, #A, print(A[n,])) }

Formula

T(n,k) = n!*A362826(n,k) for k > 0.

A362819 Number of ordered pairs of involutions on [n] that commute.

Original entry on oeis.org

1, 1, 4, 10, 52, 196, 1216, 5944, 42400, 250912, 2008576, 13815616, 122074624, 950640640, 9158267392, 79258479616, 824644235776, 7823203807744, 87245790791680, 897748312609792, 10665239974537216, 118040852776093696, 1486172381689544704, 17572063073426206720, 233446797379437248512
Offset: 0

Views

Author

Andrew Howroyd, May 05 2023

Keywords

Comments

Two involutions x,y on [n] commute if x*y = y*x (i.e. x(y(i)) = y(x(i)) for i in [n]).

Links

Crossrefs

Column k=2 of A362824.
A053529 is the corresponding sequence for all permutations.
Cf. A000085, A000898, A181162, A362820, A362825.

Programs

  • PARI
    b(n,f) = {sum(k=0, n\2, f(k)*binomial(n,2*k)*(2*k)!/(k!*2^k))}
a(n) = {b(n, k->b(n-2*k, j->1)*b(k, j->2^(k-j)))}
  • PARI
    seq(n)=Vec(serlaplace(exp(x + 3*x^2/2 + x^4/4 + O(x*x^n))))

Formula

a(n) = Sum_{k=0..floor(n/2)} A000085(n-2*k) * A000898(k) * binomial(n,2*k) * (2*k)! / (k!*2^k).
E.g.f.: exp(x + 3*x^2/2 + x^4/4).

A362823 Number of labeled magmas with n elements satisfying the identities (xy)y = x and (xy)z = (xz)y.

Original entry on oeis.org

1, 1, 4, 22, 976, 19376, 7680016, 430723168, 1489656111616, 214815786486400, 6364561150037368576, 2241692646969785651456, 566719960584895502028138496, 471612192582034433034750951424, 1008512943343839231897776246546624512, 1936475539456937172034340659334701398016
Offset: 0

Views

Author

Andrew Howroyd, May 06 2023

Keywords

Comments

a(n) is the number of ordered n-tuples of involutions on [n] that pairwise commute. Two involutions x,y on [n] commute if x*y = y*x.

Links

Crossrefs

Main diagonal of A362824.
Cf. A362383, A362643, A362821, A362822 (isomorphism classes).

A362821 Number of labeled right involutory Płonka magmas with n elements.

Original entry on oeis.org

1, 1, 2, 10, 70, 916, 16636, 494824, 20486432, 1320568624, 119526633136, 16466317431136, 3210471529800448, 940911157770908416, 392559353168479443584, 244017910517578226511616, 216775189886094655708439296, 284778404550532041821957456896, 536018349047631899870416803595264
Offset: 0

Views

Author

Andrew Howroyd, May 08 2023

Keywords

Comments

A right involutory Płonka magma is a magma satisfying the identities (xy)y = x, (xy)z = (xz)y and x(yz) = xy. See A361720 for additional information.

Links

Crossrefs

Cf. A361720 (isomorphism classes), A362643, A362822, A362824.

Programs

  • PARI
    \\ D(n,k) is e.g.f. of column k of A362824.
B(n, k)={polcoef(x^k/prod(j=0, k, 1-2^j*x + O(x*x^n)), n)}
D(n, k)=exp(sum(j=0, min(k, logint(n, 2)), B(k, j)*x^(2^j)/2^j, O(x*x^n)))
seq(n)=Vec(serlaplace(sum(k=0, n, log(D(n-k+1, k))^k/k!)))

Formula

E.g.f.: Sum_{k>=0} log(B(k,x))^k/k! where B(k,x) is the e.g.f. of column k of A362824.

A362825 Number of ordered triples of involutions on [n] that pairwise commute.

Original entry on oeis.org

1, 1, 8, 22, 232, 1016, 12496, 73648, 1032032, 7586272, 118141696, 1033672256, 17668427008, 178649596672, 3313667912192, 37898019913216, 756948065453056, 9640771045925888, 205935949714235392, 2885307792776353792, 65568056040976818176
Offset: 0

Views

Author

Andrew Howroyd, May 06 2023

Keywords

Comments

Two involutions x,y on [n] commute if x*y = y*x.

Links

Crossrefs

Column k=3 of A362824.
Cf. A362819.

Programs

  • PARI
    seq(n) = {Vec(serlaplace(exp(x + 7*x^2/2 + 7*x^4/4 + x^8/8 + O(x*x^n))))}

Formula

E.g.f.: exp(x + 7*x^2/2 + 7*x^4/4 + x^8/8).

A362903 Array read by antidiagonals: T(n,k) is the number of nonisomorphic k-tuples of involutions on a (2n)-set that pairwise commute.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 11, 4, 1, 1, 16, 43, 24, 5, 1, 1, 32, 171, 176, 46, 6, 1, 1, 64, 683, 1376, 611, 80, 7, 1, 1, 128, 2731, 10944, 9281, 1864, 130, 8, 1, 1, 256, 10923, 87424, 146445, 54384, 5161, 200, 9, 1, 1, 512, 43691, 699136, 2334181, 1696352, 285939, 13184, 295, 10, 1
Offset: 0

Views

Author

Andrew Howroyd, May 11 2023

Keywords

Comments

Two involutions x,y commute if x*y = y*x. Isomorphism is up to permutation of the elements of the (2n)-set. T(n,k) also gives the values for a (2n+1)-set.

Examples

			Array begins:
======================================================
n/k| 0 1   2     3       4          5            6 ...
---+--------------------------------------------------
0  | 1 1   1     1       1          1            1 ...
1  | 1 2   4     8      16         32           64 ...
2  | 1 3  11    43     171        683         2731 ...
3  | 1 4  24   176    1376      10944        87424 ...
4  | 1 5  46   611    9281     146445      2334181 ...
5  | 1 6  80  1864   54384    1696352     53885632 ...
6  | 1 7 130  5161  285939   17562679   1110290303 ...
7  | 1 8 200 13184 1372224  165343616  20774749952 ...
8  | 1 9 295 31532 6101080 1436647664 358238974304 ...
  ...

Links

Crossrefs

Columns k=0..3 are A000012, A000027(n-1), A001752, A362904.
Rows n=1..3 are A000079, A007583, A103334(n+1).
Cf. A022166, A362648, A362824, A362826.

Programs

  • PARI
    \\ B(n, k) is A022166.
B(n, k)={polcoef(x^k/prod(j=0, k, 1-2^j*x + O(x*x^n)), n)}
C(k,n) = Vec(1/prod(j=0, min(k-1, logint(n, 2)), (1 - x^(2^j) + O(x*x^n))^B(k,j+1), 1 - x + O(x*x^n)))
M(n,m=n) = Mat(vector(m+1, k, C(k-1, n)~))
{ my(A=M(7)); for(i=1, #A, print(A[i,])) }

Formula

G.f. of column k: 1/((1 - x)*Product_{j=0..k-1} (1 - x^(2^j))^A022166(k,j+1)).
Showing 1-6 of 6 results.

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