A209667 -id:A209667 - cp's OEIS frontend

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.

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

Wouter Meeussen, Mar 11 2012

nonn

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

Alois P. Heinz, Table of n, a(n) for n = 0..500
Wikipedia, Symmetric Polynomials
FindStat - Combinatorial Statistic Finder, Semistandard Young tableaux

Cf. A191714, A209664, A209665, A209666, A209667, A209668, A209669, A209670, A209671, A209672, A209673.

Main diagonal of A210391. - Alois P. Heinz, Mar 22 2012

(* 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 *)

a(12)-a(22) from Alois P. Heinz, Mar 11 2012

Typo in Mathematica program fixed by Vaclav Kotesovec, Mar 19 2015

A209666 T(n,k) = count of degree k monomials in the complete homogeneous symmetric polynomials h(mu,k) summed over all partitions mu of n.

Original entry on oeis.org

1, 2, 7, 3, 18, 55, 5, 50, 216, 631, 7, 118, 729, 2780, 8001, 11, 301, 2621, 12954, 45865, 130453, 15, 684, 8535, 55196, 241870, 820554, 2323483, 22, 1621, 28689, 241634, 1307055, 5280204, 17353028, 48916087, 30, 3620, 91749, 1012196, 6783210, 32711022, 124991685, 401709720, 1129559068

Offset: 1

Wouter Meeussen, Mar 11 2012

			Table starts as:
  1;
  2,   7;
  3,  18, 55;
  5,  50, 216,  631;
  7, 118, 729, 2780, 8001;

Alois P. Heinz, Rows n = 1..141, flattened
Wikipedia, Symmetric Polynomials

Main diagonal is A209668; row sums are A209667.

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:
T:= (n, k)-> b(n$2, k):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Mar 04 2016

h[n_, v_] := Tr@ Apply[Times, Table[Subscript[x, j], {j, v}]^# & /@ Compositions[n, v], {1}]; h[par_?PartitionQ, v_] := Times @@ (h[#, v] & /@ par); Table[Tr[(h[#, k] & /@ Partitions[l]) /. Subscript[x, _] -> 1], {l, 10}, {k, l}]

A209768 Triangle of coefficients of polynomials v(n,x) jointly generated with A209767; see the Formula section.

Original entry on oeis.org

1, 2, 3, 3, 7, 7, 4, 14, 26, 17, 5, 24, 64, 83, 41, 6, 37, 130, 251, 250, 99, 7, 53, 233, 599, 899, 723, 239, 8, 72, 382, 1232, 2478, 3022, 2034, 577, 9, 94, 586, 2282, 5774, 9476, 9700, 5607, 1393, 10, 119, 854, 3908, 11952, 24734, 34152, 30063

Offset: 1

Clark Kimberling, Mar 15 2012

For a discussion and guide to related arrays, see A208510.

			First five rows:
1
2...3
3...7....7
4...14...26...17
5...24...64...83...41
First three polynomials v(n,x): 1, 2 + 3x , 3 + 7x + 7x^2.

Cf. A209667, A208510.

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_] := 2 x*u[n - 1, x] + (x + 1)*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[%]    (* A209767 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209768 *)

u(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x),
v(n,x)=2x*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

cp's OEIS Frontend

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.

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A209666 T(n,k) = count of degree k monomials in the complete homogeneous symmetric polynomials h(mu,k) summed over all partitions mu of n.

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A209768 Triangle of coefficients of polynomials v(n,x) jointly generated with A209767; see the Formula section.

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