A209673 -id:A209673 - cp's OEIS frontend

A210391 Number A(n,k) of semistandard Young tableaux over all partitions of n with maximal element <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 9, 6, 1, 0, 1, 5, 16, 19, 9, 1, 0, 1, 6, 25, 44, 39, 12, 1, 0, 1, 7, 36, 85, 116, 69, 16, 1, 0, 1, 8, 49, 146, 275, 260, 119, 20, 1, 0, 1, 9, 64, 231, 561, 751, 560, 189, 25, 1, 0

Offset: 0

Alois P. Heinz, Mar 20 2012

			Square array A(n,k) begins:
  1,  1,   1,   1,   1,    1,    1, ...
  0,  1,   2,   3,   4,    5,    6, ...
  0,  1,   4,   9,  16,   25,   36, ...
  0,  1,   6,  19,  44,   85,  146, ...
  0,  1,   9,  39, 116,  275,  561, ...
  0,  1,  12,  69, 260,  751, 1812, ...
  0,  1,  16, 119, 560, 1955, 5552, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
FindStat - Combinatorial Statistic Finder, Semistandard Young tableaux
Wikipedia, Young tableau

Rows n=0-10 give: A000012, A001477, A000290, A005900, A139594, A210427, A210428, A210429, A210430, A210431, A210432.
Columns k=0-8 give: A000007, A000012, A002620(n+2), A038163, A054498, A181477, A181478, A181479, A181480.
Main diagonal gives: A209673.
Cf. A138177, A191714.

# First program:
h:= (l, k)-> mul(mul((k+j-i)/(1+l[i] -j +add(`if`(l[t]>=j, 1, 0)
                 , t=i+1..nops(l))), j=1..l[i]), i=1..nops(l)):
g:= proc(n, i, k, l)
      `if`(n=0, h(l, k), `if`(i<1, 0, g(n, i-1, k, l)+
      `if`(i>n, 0, g(n-i, i, k, [l[], i]))))
    end:
A:= (n, k)-> `if`(n=0, 1, g(n, n, k, [])):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second program:
gf:= k-> 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)):
A:= (n, k)-> coeff(series(gf(k), x, n+1), x, n):
seq(seq(A(n, d-n), n=0..d), d=0..12);

(* First program: *)
h[l_, k_] := Product[Product[(k+j-i)/(1+l[[i]]-j + Sum[If[l[[t]] >= j, 1, 0], {t, i+1, Length[l]}]), {j, 1, l[[i]]}], {i, 1, Length[l]}]; g [n_, i_, k_, l_] := If[n == 0, h[l, k], If[i < 1, 0, g[n, i-1, k, l] + If[i > n, 0, g[n-i, i, k, Append[l, i]]]]]; a[n_, k_] := If[n == 0, 1, g[n, n, k, {}]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten
(* second program: *)
gf[k_] := 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)); a[n_, k_] := Coefficient[Series[gf[k], {x, 0, n+1}], x, n]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 09 2013, translated from Maple *)

G.f. of column k: 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)).

A(n,k) = Sum_{i=0..k} C(k,i) * A138177(n,k-i). - Alois P. Heinz, Apr 06 2015

A124577 Define p(alpha) to be the number of H-conjugacy classes where H is a Young subgroup of type alpha of the symmetric group S_n. Then a(n) = sum p(alpha) where |alpha| = n and alpha has at most n parts.

Original entry on oeis.org

1, 1, 6, 39, 356, 4055, 57786, 983535, 19520264, 441967518, 11235798510, 316719689506, 9800860032876, 330230585628437, 12032866998445818, 471416196117401340, 19758835313514076176, 882185444649249777913, 41797472220815112375966, 2094455101139881670407954

Offset: 0

Richard Bayley (r.t.bayley(AT)qmul.ac.uk), Nov 05 2006

nonn

p((0,n)) = A000041, p((1,n)) = A000070, p((2,n)) = A093695;
Also main diagonal of A209664. - Wouter Meeussen, Mar 11 2012
Number of partitions of n into n sorts of parts. a(2) = 6: [2a], [2b], [1a,1a], [1a,1b], [1b,1a], [1b,1b]. - Alois P. Heinz, Sep 08 2014

			E.g p((2,1)) = # H-conjugacy classes of S_3 where H = Yng((2,1)) isom S_2 times S_1 . Then a(3) = p((3)) + p((2,1)) + p((2,0,1)) + p((1,2)) + p((1,1,1))+ p((1,0,2)+ p((0,2,1)) + p((0,1,2)) + p((0,0,3)) = 3+4+4+4+6+4+3+4+4+3 = 39.

Alois P. Heinz, Table of n, a(n) for n = 0..250
Richard Bayley, Homepage.
Richard Bayley, Relative Character Theory and the Hyperoctahedral Group, Ph.D. thesis, Queen Mary College, University of London, to be published 2007.
Steve Donkin, Invariant functions on Matrices, Math. Proc. Camb. Phil. Soc. 113 (1993) 23-43.
Wikipedia, Symmetric Polynomials

Cf. A124578, A000041, A000070, A093695, A209664-A209673, A291698.

Main diagonal of A246935.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 08 2014

p[n_Integer, v_] := Sum[Subscript[x, j]^n, {j, v}];
p[par_List, v_] := Times @@ (p[#, v] & /@ par);
Tr /@ Table[(p[#, l] & /@ IntegerPartitions[l]) /. Subscript[x, ] -> 1, {l, 19}] (* _Wouter Meeussen, Mar 11 2012 *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, k*b[n - i, i, k]]]]; a[n_] := b[n, n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

{a(n)=polcoeff(1/prod(k=1,n,1-n*x^k +x*O(x^n)),n)} \\ Paul D. Hanna, Nov 26 2009

Let x = x_1x_2x_3... and x^alpha = x_1^(alpha_1)x_2^(alpha_2)x_3^(alpha_3).... Let Phi = set of all primitive necklaces. If b is a primitive necklace then C(b) = Content(b) = (beta_1, beta_2,beta_3,.....) where beta_i = the number of times i occurs in b. For example if b=[11233] then C(b) = (2,1,2). To generate the p(alpha) we do the following. sum_alpha p(alpha)x^alpha = prod_(b in Phi) prod_(k = 1)^infinity 1/(1- x^(c(b) times k )) = prod_(b in Phi) prod_(k = 1)^infinity (1+ x^(k times C(b)) + x^(2k times C(b)) + x^(3k times C(b)) + ....)
From Paul D. Hanna, Nov 26 2009: (Start)
a(n) = [x^n] Product_{k>=1} 1/(1 - n*x^k) for n>0.
a(n) = Sum_{k=1..n} A008284(n,k)*n^k, where A008284(n,k) = number of partitions of n in which the greatest part is k, 1<=k<=n. (End)
a(n) ~ n^n * (1 + 1/n + 2/n^2 + 3/n^3 + 5/n^4 + 7/n^5 + 11/n^6 + 15/n^7 + 22/n^8 + 30/n^9 + 42/n^10), where the coefficients are A000041(k)/n^k. - Vaclav Kotesovec, Mar 19 2015

Extended with formula by Paul D. Hanna, Nov 26 2009

a(0) inserted and more terms from Alois P. Heinz, Sep 08 2014

A209668 a(n) = count of monomials, of degree k = n, in the complete homogeneous symmetric polynomials h(mu,k) summed over all partitions mu of n.

Original entry on oeis.org

1, 1, 7, 55, 631, 8001, 130453, 2323483, 48916087, 1129559068, 29442232007, 835245785452, 26113646252773, 880685234758941, 32191922753658129, 1259701078978200555, 52802268925363689079, 2352843030410455053891, 111343906794849929711260, 5567596199767400904172045

Offset: 0

Wouter Meeussen, Mar 11 2012

nonn

a(n) is the number of partitions of n where each part i is marked with a word of length i over an n-ary alphabet whose letters appear in alphabetical order. a(2) = 7: 2aa, 2ab, 2bb, 1a1a, 1a1b, 1b1a, 1b1b. - Alois P. Heinz, Aug 30 2015

Alois P. Heinz, Table of n, a(n) for n = 0..386
Wikipedia, Symmetric Polynomials

Cf. A209664-A209673.
Main diagonal of A209666 and A261718.
Cf. A261783.

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-> b(n$3):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 29 2015

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[(h[#, l] & /@ Partitions[l]) /. Subscript[x, _] -> 1, {l, 10}]
b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[b[n-i*j, i-1, k] * Binomial[i+k-1, k-1]^j, {j, 0, n/i}]]]; a[n_] := b[n, n, n]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 15 2016, after Alois P. Heinz *)

a(n) ~ c * n^n, where c = A247551 = Product_{k>=2} 1/(1-1/k!) = 2.529477472... . - Vaclav Kotesovec, Nov 15 2016

a(n) = [x^n] Product_{k>=1} 1 / (1 - binomial(k+n-1,n-1)*x^k). - Ilya Gutkovskiy, May 09 2021

a(0)=1 prepended and a(11)-a(19) added by Alois P. Heinz, Aug 29 2015

A191714 a(n,k) equals the number of semistandard Young tableaux with shape of a partition of n and maximal element <= k.

Original entry on oeis.org

1, 1, 4, 1, 6, 19, 1, 9, 39, 116, 1, 12, 69, 260, 751, 1, 16, 119, 560, 1955, 5552, 1, 20, 189, 1100, 4615, 15372, 43219, 1, 25, 294, 2090, 10460, 40677, 131131, 366088, 1, 30, 434, 3740, 22220, 100562, 370909, 1168008, 3245311, 1, 36, 630, 6512, 45628, 239316, 1007083, 3570240, 11042199, 30569012, 1, 42, 882, 10868, 89420, 541926, 2596573, 10347864, 35587071, 108535130, 299662672, 1, 49, 1218, 17732, 170340, 1188341, 6466159, 28915056, 110426979, 370661885, 1117689232, 3079276708

Offset: 1

Wouter Meeussen, Jun 12 2011

Maximal element can be any integer, but is chosen here to be <=n.

			For n=3 and k=2 the SSYT are
par= {3}     SSYT= {{1, 1, 1}}, {{2, 1, 1}}, {{2, 2, 1}}, {{2, 2, 2}}
par= {2,1}   SSYT= {{2, 1}, {1}}, {{2, 2}, {1}}
par= {1,1,1} SSYT= none
counts 4+2+0 = 6 = a(3,2).
Table begins:
  1;
  1,  4;
  1,  6,  19;
  1,  9,  39,  116;
  1, 12,  69,  260,  751;
  1, 16, 119,  560, 1955,  5552;
  1, 20, 189, 1100, 4615, 15372, 43219; ...

Alois P. Heinz, Rows n = 1..44, flattened
R. Stanley, Hook Lengths and Contents

Cf. A102539, A210391.

Main diagonal gives A209673.

Needs["Combinatorica`"];
hooklength[(p_)?PartitionQ] := Block[{ferr = (PadLeft[1 + 0*Range[#1], Max[p]] &) /@ p}, DeleteCases[(Rest[FoldList[Plus, 0, #1]] &) /@ ferr + Reverse /@ Reverse[Transpose[(Rest[FoldList[Plus, 0, #1]] &) /@ Reverse[Reverse /@ Transpose[ferr]]]], 0, -1] - 1];
content[(p_)?PartitionQ]:= Block[{le= Max[p], ferr =(PadLeft[1+ 0*Range[#1], Max[p]]&) /@ p}, DeleteCases[ MapIndexed[-le+ Range[le,1,-1]- #1- Tr[#2]&, 0*ferr]*ferr,0,-1]+ le];
stanley[(p_)?PartitionQ, t_Integer] := Times @@ ((t + Flatten[content[p]])/Flatten[hooklength[p]]);
Table[Tr[ stanley[#,k]  &/@ Partitions[n] ] , {n,12}, {k,n}]

A178300 Triangle T(n,k) = binomial(n+k-1,n) read by rows, 1 <= k <= n.

Original entry on oeis.org

1, 1, 3, 1, 4, 10, 1, 5, 15, 35, 1, 6, 21, 56, 126, 1, 7, 28, 84, 210, 462, 1, 8, 36, 120, 330, 792, 1716, 1, 9, 45, 165, 495, 1287, 3003, 6435, 1, 10, 55, 220, 715, 2002, 5005, 11440, 24310, 1, 11, 66, 286, 1001, 3003, 8008, 19448, 43758, 92378, 1, 12, 78, 364, 1365, 4368, 12376, 31824, 75582, 167960, 352716, 1, 13, 91, 455, 1820, 6188, 18564, 50388, 125970, 293930, 646646, 1352078

Offset: 1

Alford Arnold, May 24 2010

Obtained from A176992 by reversing entries in each row, from A092392 by removing the left column and reversing entries in each row, or from A100100 by removing the first two columns and reversing entries in each row.
Also T(n,k) = count of degree k monomials in the Monomial symmetric polynomials m(mu,k) summed over all partitions mu of n.
T(n,k) is the number of ways to put n indistinguishable balls into k distinguishable boxes. - Dennis P. Walsh, Apr 11 2012
T(n,k) is the number of compositions of n into k parts if zeros are allowed as parts. - L. Edson Jeffery, Jul 23 2014
T(n,k) is the number of compositions (ordered partitions) of n+k into exactly k parts. - Juergen Will, Jan 23 2016
T(n,k) is the number of binary strings with exactly n zeros and k-1 ones. - Dennis P. Walsh, Apr 09 2016
T(n,k) is the number of functions f:[k-1]->[n+1] that are nondecreasing. There is a unique correspondence between such a function and a binary string with exactly n zeros and k-1 ones. Given a string, let the corresponding function f be defined by f(i)=1 + (the number of zeros in the string that precede the i-th one in the string) for i=1,..,k-1. - Dennis P. Walsh, Apr 09 2016

			Triangle begins
  1;
  1,    3;
  1,    4,   10;
  1,    5,   15,   35;
  1,    6,   21,   56,  126;
  1,    7,   28,   84,  210,  462;
  1,    8,   36,  120,  330,  792, 1716;
T(3,3)=10 since there are 10 ways to put 3 identical balls into 3 distinguishable boxes, namely, (OOO)()(), ()(OOO)(), ()()(OOO), (OO)(O)(), (OO)()(O), (O)(OO)(), ()(OO)(O), (O)()(OO), ()(O)(OO), and (O)(O)(O). - _Dennis P. Walsh_, Apr 11 2012
For example, T(3,3)=10 since there are ten functions f:[2]->[4] that are nondecreasing, namely, <f(1),f(2)> = <1,1> or <1,2> or <1,3> or <1,4> or <2,2> or <2,3> or <2,4> or <3,3> or <3,4> or <4,4>. - _Dennis P. Walsh_, Apr 09 2016

P. A. MacMahon, Memoir on the Theory of the Compositions of Numbers, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901.
Ch. Stover and E. W. Weisstein, Composition. From MathWorld - A Wolfram Web Resource.
Wikipedia, Symmetric Polynomials

Cf. A000142, A000312, A007318, A001791 (row sums), A209664-A209673.
Cf. A000217, A000292, A000332, A001700, A008292, A038675, A046899, A088218.
Cf. A176992, A092392, A100100.

// As triangle
[[Binomial(n+k-1,n): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Jan 24 2016

seq(seq(binomial(n+k-1,n),k=1..n),n=1..15); # Dennis P. Walsh, Apr 11 2012

m[par_?PartitionQ, v_] := Block[{le = Length[par], it }, If[le > v, Return[0]]; it = Permutations[PadRight[par, v]]; Tr[ Apply[Times, Table[Subscript[x, j], {j, v}]^# & /@ it, {1}]]];
Table[Tr[(m[#, k] & /@ Partitions[l]) /. Subscript[x, ] -> 1], {l, 11}, {k, l}](* _Wouter Meeussen, Mar 11 2012 *)
Quiet[Needs["Combinatorica`"], All]; Grid[Table[Length[Combinatorica`Compositions[n, k]], {n, 10}, {k, n}]] (* L. Edson Jeffery, Jul 24 2014 *)
t[n_, k_] := Binomial[n + k - 1, n]; Table[ t[n, k], {n, 10}, {k, n}] // Flatten (* Robert G. Wilson v, Jul 24 2014 *)

T(n,k) = A046899(n,k-1) = A038675(n,k)/A008292(n,k).
T(n,1) = 1.
T(n,2) = n+1.
T(n,3) = A000217(n+1).
T(n,4) = A000292(n+1).
T(n,5) = A000332(n+4).
T(n,n) = A001700(n-1) = A088218(n). - Dennis P. Walsh, Apr 10 2012

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

Wouter Meeussen, Mar 11 2012

nonn

Peter J. Taylor, Table of n, a(n) for n = 1..100
Wikipedia, Symmetric Polynomials

Cf. A209664-A209673.

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}]

Main diagonal of triangle A209669.

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

Wouter Meeussen, Mar 11 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..250
Wikipedia, Symmetric Polynomials

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

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}]

Row sums of table A209666.

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

Wouter Meeussen, Mar 11 2012

nonn

Peter J. Taylor, Table of n, a(n) for n = 1..100
Wikipedia, Symmetric Polynomials

Cf. A209664-A209673.

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}]

Row sums of triangle A209669.

More terms from Peter J. Taylor, Mar 02 2017

A209774 Triangle of coefficients of polynomials v(n,x) jointly generated with A209773; see the Formula section.

Original entry on oeis.org

1, 2, 3, 2, 7, 8, 3, 12, 25, 21, 3, 19, 56, 84, 55, 4, 26, 103, 227, 269, 144, 4, 36, 169, 486, 848, 833, 377, 5, 45, 259, 914, 2078, 2999, 2518, 987, 5, 58, 372, 1565, 4393, 8277, 10192, 7475, 2584, 6, 69, 518, 2503, 8342, 19420, 31269, 33600, 21881

Offset: 1

Clark Kimberling, Mar 15 2012

Last term in row n: F(2n), where F=A000045, the Fibonacci numbers

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

			First five rows:
1
2...3
2...7....8
3...12...25...21
3...19...56...84...55
First three polynomials v(n,x): 1, 2 + 3x , 2 + 7x + 8x^2.

Cf. A209673, 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_] := (x + 1)*u[n - 1, x] + 2 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[%]    (* A209773 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209774 *)

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

A209672 a(n) = count of monomials, of degrees k=1 to n, in the Schur symmetric polynomials s(mu,k) summed over all partitions mu of n.

Original entry on oeis.org

1, 5, 26, 165, 1093, 8203, 64516, 550766, 4911215, 46480657, 457372449, 4714813700, 50312993309, 557792410015, 6377533006104, 75321602836350, 914532538185703, 11422271100356431, 146273502952981364, 1920759273853147991, 25802956138975439144, 354546559878617075950

Offset: 1

Wouter Meeussen, Mar 11 2012

nonn

Row sums of triangle A191714.

Wikipedia, Symmetric Polynomials

Cf. A209664-A209673.

(* see A191714 *)
Tr /@ Table[Tr[stanley[#, k] & /@ Partitions[n]], {n, 10}, {k, n}]

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

cp's OEIS Frontend

A210391 Number A(n,k) of semistandard Young tableaux over all partitions of n with maximal element <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A124577 Define p(alpha) to be the number of H-conjugacy classes where H is a Young subgroup of type alpha of the symmetric group S_n. Then a(n) = sum p(alpha) where |alpha| = n and alpha has at most n parts.

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A209668 a(n) = count of monomials, of degree k = n, in the complete homogeneous symmetric polynomials h(mu,k) summed over all partitions mu of n.

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A191714 a(n,k) equals the number of semistandard Young tableaux with shape of a partition of n and maximal element <= k.

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A178300 Triangle T(n,k) = binomial(n+k-1,n) read by rows, 1 <= k <= n.

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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.

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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.

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