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 1-5 of 5 results.

A209673 a(n) = count of monomials, of degree k=n, in the Schur symmetric polynomials s(mu,k) summed over all partitions mu of n.

Original entry on oeis.org

1, 1, 4, 19, 116, 751, 5552, 43219, 366088, 3245311, 30569012, 299662672, 3079276708, 32773002718, 362512238272, 4136737592323, 48773665308176, 591313968267151, 7375591544495636, 94340754464144215, 1237506718985945656, 16608519982801477908, 228013066931927465872
Offset: 0

Views

Author

Wouter Meeussen, Mar 11 2012

Keywords

Comments

Main diagonal of triangle A191714.
a(n) is also the number of semistandard Young tableaux of size and maximal entry n. - Christian Stump, Oct 09 2015

Links

Crossrefs

Cf. A191714, A209664, A209665, A209666, A209667, A209668, A209669, A209670, A209671, A209672, A209673.
Main diagonal of A210391. - Alois P. Heinz, Mar 22 2012

Programs

  • Mathematica
    (* see A191714 *)
Tr /@ Table[(stanley[#, l] & /@ Partitions[l]), {l, 11}]
(* or *)
Table[SeriesCoefficient[1/((1-x)^(n*(n+1)/2) * (1+x)^(n*(n-1)/2)), {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Aug 06 2025 *)

Extensions

a(12)-a(22) from Alois P. Heinz, Mar 11 2012
Typo in Mathematica program fixed by Vaclav Kotesovec, Mar 19 2015

A209671 a(n) = count of monomials, of degree k=n, in the elementary symmetric polynomials e(mu,k) summed over all partitions mu of n.

Original entry on oeis.org

1, 5, 37, 405, 5251, 84893, 1556535, 33175957, 785671039, 20841132255, 604829604655, 19236214748061, 661348833658423, 24554370466786319, 976242978063976162, 41477168810872793493, 1872694395510428040983, 89644070894632864643651, 4531712537608857605836563
Offset: 1

Views

Author

Wouter Meeussen, Mar 11 2012

Keywords

Links

Crossrefs

Cf. A209664-A209673.

Programs

  • Mathematica
    e[n_, v_] := Tr[Times @@@ Select[Subsets[Table[Subscript[x, j], {j, v}]], Length[#] == n &]]; e[par_?PartitionQ, v_] := Times @@ (e[#, v] & /@ par); Tr /@ Table[(e[#, l] & /@ Partitions[l]) /. Subscript[x, _] -> 1, {l, 10}]

Formula

Main diagonal of triangle A209669.

Extensions

More terms from Peter J. Taylor, Mar 02 2017

A209667 a(n) = count of monomials, of degrees k=0 to n, in the complete homogeneous symmetric polynomials h(mu,k) summed over all partitions mu of n.

Original entry on oeis.org

1, 1, 9, 76, 902, 11635, 192205, 3450337, 73128340, 1696862300, 44414258862, 1264163699189, 39640715859359, 1340191402045395, 49097854149726795, 1924982506686743639, 80831323253459088871, 3607487926962810556542, 170964537623741430399076
Offset: 0

Views

Author

Wouter Meeussen, Mar 11 2012

Keywords

Links

Crossrefs

Cf. A209664, A209665, A209666, A209667, A209668, A209669, A209670, A209671, A209672, A209673.

Programs

  • Maple
    b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)*binomial(i+k-1, k-1)^j, j=0..n/i)))
    end:
a:= n-> add(b(n$2, k), k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 04 2016
  • Mathematica
    h[n_, v_] := Tr@ Apply[Times, Table[Subscript[x, j], {j, v}]^# & /@ Compositions[n, v], {1}]; h[par_?PartitionQ, v_] := Times @@ (h[#, v] & /@ par); Tr/@ Table[Tr[(h[#, k] & /@ Partitions[l]) /. Subscript[x, _] -> 1], {l, 10}, {k, l}]

Formula

Row sums of table A209666.

Extensions

a(0), a(11)-a(18) from Alois P. Heinz, Mar 04 2016

A209670 a(n) = count of monomials, of degrees k=1 to n, in the elementary symmetric polynomials e(mu,k) summed over all partitions mu of n.

Original entry on oeis.org

1, 6, 48, 547, 7301, 120315, 2239803, 48278809, 1153934735, 30834749017, 900390736548, 28782727026031, 993911439932097, 37039780178206877, 1477457354215115765, 62950691931099382408, 2849385291187650049208, 136701569959985165325989, 6924379544998951633495956
Offset: 1

Views

Author

Wouter Meeussen, Mar 11 2012

Keywords

Links

Crossrefs

Cf. A209664-A209673.

Programs

  • Mathematica
    e[n_, v_] := Tr[Times @@@ Select[Subsets[Table[Subscript[x, j], {j, v}]], Length[#] == n &]]; e[par_?PartitionQ, v_] := Times @@ (e[#, v] & /@ par); Tr/@ Table[Tr[(e[#, k] & /@ Partitions[l]) /. Subscript[x, _] -> 1], {l, 10}, {k, l}]

Formula

Row sums of triangle A209669.

Extensions

More terms from Peter J. Taylor, Mar 02 2017

A209770 Triangle of coefficients of polynomials v(n,x) jointly generated with A209769; see the Formula section.

Original entry on oeis.org

1, 3, 1, 5, 4, 2, 9, 12, 10, 3, 15, 29, 33, 19, 5, 25, 64, 93, 77, 37, 8, 41, 132, 234, 251, 171, 69, 13, 67, 261, 548, 719, 629, 362, 127, 21, 109, 500, 1216, 1884, 2004, 1482, 742, 230, 34, 177, 936, 2592, 4628, 5784, 5196, 3342, 1482, 412, 55, 287
Offset: 1

Views

Author

Clark Kimberling, Mar 15 2012

Keywords

Comments

Column 1: A001595
Row n ends with F(n), where F=A000045, the Fibonacci numbers.
Row sums: 1,4,11,34,101,304,911,2734,... A060925
Alternating row sums: 1,2,3,4,5,6,7,.... A000027
For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
3....1
5....4....2
9....12...10...3
15...29...33...19...5
First three polynomials v(n,x): 1, 3 + x , 5 + 4x + 2x^2.

Crossrefs

Cf. A209669, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A209769 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209770 *)

Formula

u(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x),
v(n,x)=(x+1)*u(n-1,x)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.