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
Cf.
A191714,
A209664,
A209665,
A209666,
A209667,
A209668,
A209669,
A209670,
A209671,
A209672,
A209673.
-
(* 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 *)
A209669
T(n,k) = count of degree k monomials in the elementary symmetric polynomials e(mu,k) summed over all partitions mu of n.
Original entry on oeis.org
1, 1, 5, 1, 10, 37, 1, 21, 120, 405, 1, 42, 363, 1644, 5251, 1, 85, 1117, 6814, 27405, 84893, 1, 170, 3360, 27404, 138085, 514248, 1556535, 1, 341, 10164, 111045, 701960, 3145848, 11133493, 33175957, 1, 682, 30520, 445132, 3521405, 18956548, 78337448
Offset: 1
Table starts as
1;
1, 5;
1, 10, 37;
1, 21, 120, 405;
1, 42, 363, 1644, 5251;
...
For n = 2, k = 2 the partitions of n are 2 and 1+1, which correspond respectively to (xy) contributing 1 and (x+y)*(x+y) contributing 4 for a total of 5. - _Peter J. Taylor_, Mar 01 2017
-
e[n_, v_] := Tr[Times @@@ Select[Subsets[Table[Subscript[x, j], {j, v}]], Length[#] == n &]]; e[par_?PartitionQ, v_] := Times @@ (e[#, v] & /@ par); Table[Tr[(e[#, k] & /@ Partitions[l]) /. Subscript[x, _] -> 1], {l, 10}, {k, l}]
-
# See Taylor link
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
-
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
-
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}]
Showing 1-3 of 3 results.
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