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 1-7 of 7 results.

A296188 Number of normal semistandard Young tableaux whose shape is the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 8, 1, 6, 12, 16, 6, 32, 32, 28, 1, 64, 16, 128, 24, 96, 80, 256, 8, 44, 192, 22, 80, 512, 96, 1024, 1, 288, 448, 224, 30, 2048, 1024, 800, 40, 4096, 400, 8192, 240, 168, 2304, 16384, 10, 360, 204, 2112, 672, 32768, 68, 832, 160, 5376, 5120
Offset: 1

Views

Author

Gus Wiseman, Feb 14 2018

Keywords

Comments

A tableau is normal if its entries span an initial interval of positive integers. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The a(9) = 6 tableaux:
1 3   1 2   1 2   1 2   1 1   1 1
2 4   3 4   3 3   2 3   2 3   2 2

References

  • Richard P. Stanley, Enumerative Combinatorics Volume 2, Cambridge University Press, 1999, Chapter 7.10.

Links

Crossrefs

Cf. A000085, A001222, A056239, A063834, A112798, A122111, A138178, A153452, A191714, A210391, A228125, A296150, A296560, A296561, A299202, A299966, A300056, A300121.

Programs

  • Mathematica
    conj[y_List]:=If[Length[y]===0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
conj[n_Integer]:=Times@@Prime/@conj[If[n===1,{},Join@@Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]];
ssyt[n_]:=If[n===1,1,Sum[ssyt[n/q*Times@@Cases[FactorInteger[q],{p_,k_}:>If[p===2,1,NextPrime[p,-1]^k]]],{q,Rest[Divisors[n]]}]];
Table[ssyt[conj[n]],{n,50}]

Formula

Let b(n) = Sum_{d|n, d>1} b(n * d' / d) where if d = Product_i prime(s_i)^m(i) then d' = Product_i prime(s_i - 1)^m(i) and prime(0) = 1. Then a(n) = b(conj(n)) where conj = A122111.

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.

Original entry on oeis.org

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

Views

Author

Alois P. Heinz, Mar 20 2012

Keywords

Examples

			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, ...

Links

Crossrefs

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.

Programs

  • Maple
    # 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);
  • Mathematica
    (* 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 *)

Formula

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

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

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

Views

Author

Wouter Meeussen, Mar 11 2012

Keywords

Comments

Row sums of triangle A191714.

Links

Crossrefs

Cf. A209664-A209673.

Programs

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

Extensions

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

A296560 Number of normal semistandard Young tableaux whose shape is the conjugate of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 4, 6, 6, 1, 12, 1, 8, 16, 8, 1, 28, 1, 24, 30, 10, 1, 32, 22, 12, 44, 40, 1, 96, 1, 16, 48, 14, 68, 96, 1, 16, 70, 80, 1, 220, 1, 60, 204, 18, 1, 80, 90, 168, 96, 84, 1, 224, 146, 160, 126, 20, 1, 400, 1, 22, 584, 32, 264, 416, 1, 112, 160
Offset: 1

Views

Author

Gus Wiseman, Feb 15 2018

Keywords

Comments

A tableau is normal if its entries span an initial interval of positive integers. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Crossrefs

Cf. A000085, A001222, A056239, A063834, A112798, A122111, A138178, A153452, A191714, A210391, A228125, A296150, A296188, A299202.

Programs

  • Mathematica
    a[n_]:=If[n===1,1,Sum[a[n/q*Times@@Cases[FactorInteger[q],{p_,k_}:>If[p===2,1,NextPrime[p,-1]^k]]],{q,Rest[Divisors[n]]}]];
Array[a,100]

A191716 a(n,k) equals (1/n!) multiplied by the count of permutations with cycle length k in all products u v u^-1 v^-1 over all permutations u and v of length n.

Original entry on oeis.org

1, 0, 2, 3, 0, 3, 0, 19, 0, 5, 40, 0, 73, 0, 7, 0, 492, 0, 217, 0, 11, 1260, 0, 3225, 0, 540, 0, 15, 0, 24096, 0, 14968, 0, 1234, 0, 22, 72576, 0, 232156, 0, 55594, 0, 2524, 0, 30, 0, 1922148, 0, 1524823, 0, 176800, 0, 4987, 0, 42, 6652800, 0, 24999984, 0, 7758160, 0, 496680, 0, 9120, 0, 56, 0, 227963280, 0, 216975032, 0, 32769481, 0, 1277331, 0, 16399, 0, 77
Offset: 1

Views

Author

Wouter Meeussen, Jun 12 2011

Keywords

Comments

Row sums equal n! by definition.

Examples

			1;
0, 2;
3, 0, 3;
0, 19, 0, 5;
40, 0, 73, 0, 7;
0, 492, 0, 217, 0, 11;
1260, 0, 3225, 0, 540, 0, 15;
0, 24096, 0, 14968, 0, 1234, 0, 22;

Links

Crossrefs

Cf. A191714

Programs

  • Mathematica
    (*slow:*)
Table[Rest@CoefficientList[Apply[Plus,Flatten[Outer[ q^Length[ ToCycles[#1[[#2]][[InversePermutation[#1]]][[InversePermutation[#2]]]]] &, Permutations[w], Permutations[w], 1]]],q]/w!,{w,4}]//Expand;
(*fast:*)
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];
Table[Rest@ CoefficientList[ Apply[Plus, Apply[Times, q + Flatten[content[#]]] & /@ Partitions[ k ]] , q], {k, 12}]

A191718 a(n,k) is the count of permutations with cycle length k in the products w*w over all permutations w of length n.

Original entry on oeis.org

1, 0, 2, 2, 0, 4, 0, 14, 0, 10, 24, 0, 70, 0, 26, 0, 304, 0, 340, 0, 76, 720, 0, 2548, 0, 1540, 0, 232, 0, 13488, 0, 18956, 0, 7112, 0, 764, 40320, 0, 161936, 0, 125580, 0, 32424, 0, 2620, 0, 1011456, 0, 1648160, 0, 808248, 0, 151440, 0, 9496, 3628800
Offset: 1

Views

Author

Wouter Meeussen, Jun 12 2011

Keywords

Comments

Row sums equal n! by definition.

Links

Crossrefs

Cf. A191714.

Programs

  • Mathematica
    (* slow *)
Table[Rest@ CoefficientList[ Apply[Plus, (q^Length[ToCycles[# [[#]] ]])&  /@ Permutations[n] ] ,q],{n,6}]
(* fast, content[] see A191714 *)
Table[Rest@ CoefficientList[ Apply[Plus, NumberOfTableaux[#]Apply[Times,q+Flatten[content[#]]]& /@ Partitions[n]] ,q],{n,6}]
Showing 1-7 of 7 results.

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