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 1-10 of 12 results. Next

A060350 The sum over all subsets S of [n] of the squares of the number of permutations with descent set = S.

Original entry on oeis.org

1, 1, 2, 10, 88, 1216, 24176, 654424, 23136128, 1035227008, 57186502912, 3822411268864, 304059285928960, 28385946491599360, 3073391215118186496, 381995951933025287680, 54020316243835807039488, 8624091617045072628121600, 1543536018434416280510332928
Offset: 0

Views

Author

Mike Zabrocki, Mar 31 2001

Keywords

Comments

a(n) = number of ordered pairs of permutations of [n] such that the first has an ascent wherever the second has a descent and vice versa. For example, the pair of permutations (1243, 4123) does not qualify because they have a common ascent starting at location 2, and a(2) = 2 counts (12, 21), (21, 12). - David Callan, Sep 15 2013

Examples

			a(1)=1^2; a(2)=1^2+1^2; a(3)=1^2+2^2+2^2+1^2; a(4)=1^2+3^2+5^2+3^2+3^2+5^2+3^2+1^2.

Links

Crossrefs

Cf. A060351, A262233, A262234, A262241, A262372.
Row sums of A259465.
Column k=2 of A334622.

Programs

  • Maple
    ct := proc(k) option remember; local i,out,n; if k=0 then RETURN(1); fi; n := floor(evalf(log[2](k)))+1; if k=2^n or k=2^(n+1)-1 then RETURN(1); fi; out := 0; for i from 1 to n do if irem(iquo(k, 2^(i-1)), 2) = 1 and irem(iquo(2*k,2^(i-1)),2) =0 then out := out+(n-1)!/(i-1)!/(n-i)!* ct(floor(irem(k,2^(i-1))+2^(i-2)))*ct(iquo(k,2^i)); fi; od; out; end: seq(add(ct(i)^2,i=floor(2^(n-1))..2^n-1), n=0..15);
# second Maple program:
b:= proc(u, o, h) option remember; `if`(u+o=0, 1,
      add(add(b(u-j, o+j-1, h+i-1), i=1..u+o-h), j=1..u)+
      add(add(b(u+j-1, o-j, h-i), i=1..h), j=1..o))
    end:
a:= n-> b(0, n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 02 2015
  • Mathematica
    b[u_, o_, h_] := b[u, o, h] = If[u + o == 0, 1, Sum[Sum[b[u - j, o + j - 1, h + i - 1], {i, 1, u + o - h}], {j, 1, u}] + Sum[Sum[b[u + j - 1, o - j, h - i], {i, 1, h}], {j, 1, o}]]; a[n_] := b[0, n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

Formula

a(n) = A137782(2n) / A000984(n).
a(n) = Sum_{j=0..ceiling(2^(n-1))-1} A060351(n,j)^2. - Alois P. Heinz, Sep 15 2020
a(n) ~ c * d^n * n!^2, where d = 0.552406011965766199179395470003589240257321... and c = 1.6412834540969426814342654061364... - Vaclav Kotesovec, Sep 18 2020

Extensions

Two more terms from Max Alekseyev, May 06 2009
a(0) prepended, a(18) from Alois P. Heinz, Jul 02 2015

A262234 Number of ordered pairs (p,q) of permutations of [n] with equal up-down signatures and p(1)=q(1) if n>0.

Original entry on oeis.org

1, 1, 2, 6, 36, 376, 6040, 137320, 4188320, 164844064, 8129435904, 490812707456, 35601674684160, 3054648986802432, 305980722047302144, 35383891435049320960, 4678354778386866393088, 701273672223926436540416, 118292442535368693562662912
Offset: 0

Views

Author

Alois P. Heinz, Sep 15 2015

Keywords

Examples

			a(1) = 1: (1,1).
a(2) = 2: (12,12), (21,21).
a(3) = 6: (123,123), (132,132), (213,213), (231,231), (312,312), (321,321).
a(4) = 36: (1234,1234), (1243,1243), (1243,1342), (1324,1324), (1324,1423), (1342,1243), (1342,1342), (1423,1324), (1423,1423), (1432,1432), (2134,2134), (2143,2143), (2314,2314), (2314,2413), (2341,2341), (2413,2314), (2413,2413), (2431,2431), (3124,3124), (3142,3142), (3142,3241), (3214,3214), (3241,3142), (3241,3241), (3412,3412), (3421,3421), (4123,4123), (4132,4132), (4132,4231), (4213,4213), (4213,4312), (4231,4132), (4231,4231), (4312,4213), (4312,4312), (4321,4321).

Links

Crossrefs

Row sums of A262372.
Cf. A060350, A262233, A262241.

Programs

  • Maple
    b:= proc(u, o, h) option remember; `if`(u+o=0, 1,
      add(add(b(u-j, o+j-1, h+i-1), i=1..u+o-h), j=1..u)+
      add(add(b(u+j-1, o-j, h-i), i=1..h), j=1..o))
    end:
a:= n-> `if`(n=0, 1, add(b(j-1, n-j, n-j), j=1..n)):
seq(a(n), n=0..20);
  • Mathematica
    b[u_, o_, h_] := b[u, o, h] = If[u + o == 0, 1, Sum[Sum[b[u - j, o + j - 1, h + i - 1], {i, 1, u + o - h}], {j, 1, u}] + Sum[Sum[b[u + j - 1, o - j, h - i], {i, 1, h}], {j, 1, o}]];
a[n_] := If[n == 0, 1, Sum[b[j - 1, n - j, n - j], {j, 1, n}]];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 19 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n * n!^2 / n, where d = 0.552406011965766199179395470003589240257321... and c = 2.1899604476932970295731699544312... - Vaclav Kotesovec, Sep 18 2020

A262379 Number of ordered pairs (p,q) of permutations of [2n] with equal up-down signatures and p(1)=q(1)=n.

Original entry on oeis.org

1, 1, 8, 852, 438496, 678914816, 2475764410944, 18237517555977472, 244043425473888612352, 5486719044572824902107136, 195206678980725195413273903104, 10481263341014180286866656598294528, 817228517264548077840269973629276061696
Offset: 0

Views

Author

Alois P. Heinz, Sep 20 2015

Keywords

Examples

			a(2) = 8: (2134,2134), (2143,2143), (2314,2314), (2314,2413), (2341,2341), (2413,2314), (2413,2413), (2431,2431).

Links

Crossrefs

Cf. A262372.

Programs

  • Maple
    b:= proc(u, o, h) option remember; `if`(u+o=0, 1,
      add(add(b(u-j, o+j-1, h+i-1), i=1..u+o-h), j=1..u)+
      add(add(b(u+j-1, o-j, h-i), i=1..h), j=1..o))
    end:
a:= n-> `if`(n=0, 1, b(n-1, n, n)):
seq(a(n), n=0..15);

Formula

a(n) = A262372(2n,n).

A262479 Number of ordered pairs (p,q) of permutations of [n] with equal up-down signatures and p(1)=q(1)=2.

Original entry on oeis.org

1, 2, 8, 68, 952, 19312, 533544, 19209584, 873274112, 48904686848, 3307732742656, 265835173240320, 25041294346380800, 2732759857218994176, 342041866136488863232, 48672372916773453011968, 7813786873387535522529280, 1405550552561695118460387328
Offset: 2

Views

Author

Alois P. Heinz, Sep 30 2015

Keywords

Links

Crossrefs

Column k=2 of A262372.

Programs

  • Maple
    b:= proc(u, o, h) option remember; `if`(u+o=0, 1,
      add(add(b(u-j, o+j-1, h+i-1), i=1..u+o-h), j=1..u)+
      add(add(b(u+j-1, o-j, h-i), i=1..h), j=1..o))
    end:
a:= n-> (k-> b(k-1, n-k, n-k))(2):
seq(a(n), n=2..20);

A321059 Number of ordered pairs (p,q) of permutations of [n] with equal up-down signatures and p(1)=q(1)=3.

Original entry on oeis.org

2, 8, 64, 852, 17008, 467696, 16860608, 769534688, 43316046592, 2945846638464, 238048843712512, 22542120635302912, 2472277850712350720, 310880199760083564032, 44429722692959679590400, 7161345941202962487965696, 1292988538929694918729072640
Offset: 3

Views

Author

Alois P. Heinz, Oct 26 2018

Keywords

Examples

			a(3) = 2: (312,312), (321,321).
a(4) = 8: (3124,3124), (3142,3142), (3142,3241), (3214,3214), (3241,3142), (3241,3241), (3412,3412), (3421,3421).

Links

Crossrefs

Column k=3 of A262372.

Programs

  • Maple
    b:= proc(u, o, h) option remember; `if`(u+o=0, 1,
      add(add(b(u-j, o+j-1, h+i-1), i=1..u+o-h), j=1..u)+
      add(add(b(u+j-1, o-j, h-i), i=1..h), j=1..o))
    end:
a:= n-> (k-> b(k-1, n-k, n-k))(3):
seq(a(n), n=3..20);

A321060 Number of ordered pairs (p,q) of permutations of [n] with equal up-down signatures and p(1)=q(1)=4.

Original entry on oeis.org

10, 68, 852, 16328, 438496, 15608288, 707767328, 39729927744, 2700561836672, 218405906177792, 20715182759975936, 2276573425946788864, 286926691657095745024, 41104621803491951127552, 6641412815937114696669184, 1201977736831477123144658944
Offset: 4

Views

Author

Alois P. Heinz, Oct 26 2018

Keywords

Links

Crossrefs

Column k=4 of A262372.

Programs

  • Maple
    b:= proc(u, o, h) option remember; `if`(u+o=0, 1,
      add(add(b(u-j, o+j-1, h+i-1), i=1..u+o-h), j=1..u)+
      add(add(b(u+j-1, o-j, h-i), i=1..h), j=1..o))
    end:
a:= n-> (k-> b(k-1, n-k, n-k))(4):
seq(a(n), n=4..20);

A321066 Number of ordered pairs (p,q) of permutations of [n] with equal up-down signatures and p(1)=q(1)=10.

Original entry on oeis.org

1035227008, 48904686848, 2945846638464, 218405906177792, 19438574276705792, 2038137238577944576, 248048061418810109952, 34621419911316556017664, 5486719044572824902107136, 978934483626207520876986368, 195206678980725195413273903104
Offset: 10

Views

Author

Alois P. Heinz, Oct 26 2018

Keywords

Links

Crossrefs

Column k=10 of A262372.

Programs

  • Maple
    b:= proc(u, o, h) option remember; `if`(u+o=0, 1,
      add(add(b(u-j, o+j-1, h+i-1), i=1..u+o-h), j=1..u)+
      add(add(b(u+j-1, o-j, h-i), i=1..h), j=1..o))
    end:
a:= n-> (k-> b(k-1, n-k, n-k))(10):
seq(a(n), n=10..20);

A321061 Number of ordered pairs (p,q) of permutations of [n] with equal up-down signatures and p(1)=q(1)=5.

Original entry on oeis.org

88, 952, 17008, 438496, 15214848, 678914816, 37727416576, 2548520444480, 205352644205568, 19438574276705792, 2134484763328610304, 269000435778983860224, 38553984887490450800640, 6234291052542667315675136, 1129460021283312750345682944
Offset: 5

Views

Author

Alois P. Heinz, Oct 26 2018

Keywords

Crossrefs

Column k=5 of A262372.

A321062 Number of ordered pairs (p,q) of permutations of [n] with equal up-down signatures and p(1)=q(1)=6.

Original entry on oeis.org

1216, 19312, 467696, 15608288, 678914816, 37083546112, 2475764410944, 197890932247680, 18629724957708800, 2038137238577944576, 256240757035057154048, 36670707901174666768384, 5924904192825339158446080, 1073049649261575998090559488
Offset: 6

Views

Author

Alois P. Heinz, Oct 26 2018

Keywords

Crossrefs

Column k=6 of A262372.

A321063 Number of ordered pairs (p,q) of permutations of [n] with equal up-down signatures and p(1)=q(1)=7.

Original entry on oeis.org

24176, 533544, 16860608, 707767328, 37727416576, 2475764410944, 195463415776768, 18237517555977472, 1982309731660474368, 248048061418810109952, 35377310696428547145728, 5702015244617382235000832, 1030924606332352092323577856, 207727724955575700214833766400
Offset: 7

Views

Author

Alois P. Heinz, Oct 26 2018

Keywords

Crossrefs

Column k=7 of A262372.
Showing 1-10 of 12 results. Next

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