A053529 -id:A053529 - cp's OEIS frontend

A210590 Triangle of numbers generated by the Nekrasov-Okounkov formula.

1, 1, 1, 4, 5, 1, 18, 29, 12, 1, 120, 218, 119, 22, 1, 840, 1814, 1285, 345, 35, 1, 7920, 18144, 14674, 5205, 805, 51, 1, 75600, 196356, 185080, 79219, 16450, 1624, 70, 1, 887040, 2427312, 2515036, 1258628, 324569, 43568, 2954, 92, 1, 10886400, 32304240, 37012572, 21034376, 6431733, 1088409, 101178, 4974, 117, 1

Offset: 0

Wouter Meeussen, Mar 24 2012

Row sums are A000712, alternating sign row sums are zero (except for first row); application of the Nekrasov-Okounkov formula; see A138782.

			Table starts as:
     1;
     1,     1;
     4,     5,     1;
    18,    29,    12,    1;
   120,   218,   119,   22,   1;
   840,  1814,  1285,  345,  35,  1;
  7920, 18144, 14674, 5205, 805, 51,  1;
  ...

G. C. Greubel, Rows n=0..50 of triangle, flattened
Richard P. Stanley, Some Combinatorial Properties of Hook Lengths, Contents, and Parts of Partitions arXiv:0807.0383 [math.CO], 2009.

Cf. A000712, A053529, A057623, A138782, A234937.

T(2n,n) gives A338755.

w=9; MapIndexed[ CoefficientList[#1,t] Tr[#2-1]! &, CoefficientList[Series[Product[(1-x^i)^(-1-t), {i,w}], {x,0,w}], x]];
or alternatively:
CoefficientList[#, t] & /@ Table[1/n! Tr[(NumberOfTableaux[#1]^2 Apply[Times, t + Flatten[hooklength[#1]]^2] &) /@ Partitions[n]], {n,0,9}]
or alternatively:
Table[1/n!Tr[NumberOfTableaux[#]^2 f[ Flatten[hooklength[#]]^2,e,k,n ]&/@ Partitions[n] ],{n,0,9},{k,0,n}]
with e and f defined as:
e[n_,v_]:= Tr[Times @@@ Select[Subsets[Table[Subscript[x,j],{j,v}]],Length[#]==n&]];
f[li_List,fun_,par_,k_]:=fun[par,k]/.Thread[Array[Subscript[x,#1]&,Length[li]]->li];

E.g.f.: Product_{i=1..n} (1 - x^i)^(-1 - t).

A239841 Ordered pairs of permutation functions on n elements where f(g(g(x))) = g(g(f(x))).

Original entry on oeis.org

1, 1, 4, 30, 312, 3720, 64080, 1305360, 33949440, 1019692800, 36360576000, 1487539468800, 69633899596800, 3649476307276800, 213929162589542400, 13848506938506240000, 986705192227442688000, 76724136092268048384000, 6491471142159880740864000

Offset: 0

Chad Brewbaker, Mar 27 2014

nonn

From Paul Boddington, Feb 24 2015: (Start)
Suppose G is the symmetric group on n letters. For each g in G, the set of f satisfying fgg = ggf is just the centralizer Z_gg(G). However |Z_gg(G)| is clearly constant on conjugacy classes of G. By the orbit-stabilizer theorem the size of the conjugacy class containing g is |G| / |Z_g(G)|. Since |G| = n! and Z_g(G) is a subgroup of Z_gg(G) we see that a(n) equals n! multiplied by the sum of indices |Z_gg(G) : Z_g(G)| where the sum is over representatives of the conjugacy classes of G. Since the conjugacy classes of G correspond to partitions of n (A000041), this makes it relatively easy to find terms.
a(n) appears to equal n! * A082733(n).
(End)

John F. Humphreys, A Course In Group Theory, Oxford Science Publications, 1996, chapter 10.

Hiroaki Yamanouchi, Table of n, a(n) for n = 0..60
Eric Weisstein's World of Mathematics, Symmetric Group

Cf. A000012, A000085, A000142, A088311, A053529, A001044.

a(8)-a(9) from Giovanni Resta, Mar 27 2014

More terms from Paul Boddington, Feb 23 2015

A306042 Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1 + x)^k).

Original entry on oeis.org

1, 1, 3, 8, 50, 94, 2446, -9024, 297216, -3183264, 64191984, -1041792192, 22098943632, -478805234064, 11856288460272, -308662348027008, 8575865689645440, -248582819381690880, 7556655091130023680, -240521346554744194560, 8049494171497089265920, -283469026458500121634560

Offset: 0

Ilya Gutkovskiy, Jun 17 2018

sign

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A000041, A006252, A053529, A167137, A320349.

a:=series(mul(1/(1-log(1+x)^k),k=1..100),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 26 2019

nmax = 21; CoefficientList[Series[Product[1/(1 - Log[1 + x]^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k] Log[1 + x]^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] PartitionsP[k] k!, {k, 0, n}], {n, 0, 21}]

E.g.f.: exp(Sum_{k>=1} sigma(k)*log(1 + x)^k/k).

a(n) = Sum_{k=0..n} Stirling1(n,k)*A000041(k)*k!.

A072169 Commuting permutations: number of ordered triples of permutations f, g, h in Symm(n) which all commute.

Original entry on oeis.org

1, 1, 8, 48, 504, 4680, 66240, 856800, 14515200, 242040960, 4775500800, 95520902400, 2175146265600, 50438868480000, 1292330988748800, 34092378448128000, 971277752180736000, 28566680100102144000, 896191466580393984000, 29029508406664077312000

Offset: 0

N. J. A. Sloane, Sep 06 2003. More terms from A061256 from N. J. A. Sloane, Jun 13 2012

nonn

a(1)-a(7) computed by John McKay, Sep 06 2003.

T. D. Noe, Table of n, a(n) for n = 0..200

Column k=3 of A362827.

Cf. A061256, A053529.

for n in {1 .. 5} do G := SymmetricGroup(n); t1 := 0; for g in G do for h in G do for i in G do if g*h eq h*g and g*i eq i*g and h*i eq i*h then t1 := t1+1; end if; end for; end for; end for; n, t1; end for;

nn = 20; b = Table[DivisorSigma[1, n], {n, nn}]; Range[0, nn]! CoefficientList[Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}],  x] (* T. D. Noe, Jun 19 2012 *)

Equals A061256(n)*n!.

A239836 Number of ordered pairs of permutation functions f,g on a size n set where f(g(g(x))) = g(f(f(x))).

Original entry on oeis.org

1, 1, 2, 6, 48, 360, 2880, 20160, 241920, 3265920, 47174400, 678585600, 12933043200, 193037644800, 3661488230400, 74537438976000, 1736591560704000, 36991492521984000

Offset: 0

Chad Brewbaker, Mar 27 2014

Cf. A000012, A000085, A000142, A088311, A053529, A001044, A239779, A255525.

a(n) = A255525(n) * n!.

a(8)-a(9) from Giovanni Resta, Mar 27 2014

a(10)-a(16) from Max Alekseyev, Jan 29 2025

A305127 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^(sigma(k)/k), where sigma(k) is the sum of the divisors of k.

Original entry on oeis.org

1, 1, 5, 23, 179, 1279, 13699, 135085, 1764377, 22527521, 344625461, 5283739471, 94562354875, 1685808248383, 33947023942259, 694786150879829, 15613612524749489, 357353282848083265, 8880505496901812197, 224851013929747732231, 6106205671049245677251, 169523515381173773551871

Offset: 0

Ilya Gutkovskiy, May 26 2018

nonn

a(n)/n! is the Euler transform of [1, 3/2, 4/3, 7/4, 6/5, ... = sums of reciprocals of divisors of 1, 2, 3, 4, 5, ...].

Vaclav Kotesovec, Table of n, a(n) for n = 0..438
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.
N. J. A. Sloane, Transforms

Cf. A000203, A007429, A017665, A017666, A028342, A053529, A061256, A072169, A318769, A318814.

with(numtheory): a := proc(n) option remember; `if`(n = 0, 1, add(add(sigma(d), d = divisors(j))*a(n-j), j = 1..n)/n) end proc; seq(n!*a(n), n = 0..20); # Vaclav Kotesovec, Sep 04 2018

nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[1, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[Sum[x^(j k)/(j k (1 - x^(j k))), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d DivisorSigma[-1, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 21}]
nmax = 21; s = 1 - x; Do[s *= Sum[Binomial[DivisorSigma[1, k]/k, j]*(-1)^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 03 2018 *)

E.g.f.: Product_{k>=1} 1/(1 - x^k)^(A017665(k)/A017666(k)).
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} x^(j*k)/(j*k*(1 - x^(j*k)))).
log(a(n)/n!) ~ sqrt(n) * Pi^2 / 3. - Vaclav Kotesovec, Sep 04 2018

A319600 Number T(n,k) of plane partitions of n into parts of exactly k sorts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 3, 4, 0, 6, 22, 18, 0, 13, 96, 198, 120, 0, 24, 330, 1272, 1800, 840, 0, 48, 1146, 7518, 19152, 20640, 7920, 0, 86, 3518, 36684, 148200, 274080, 234720, 75600, 0, 160, 10946, 177438, 1080960, 3083640, 4462560, 3180240, 887040, 0, 282, 32102, 788928, 6952440, 28621920, 62056080, 73175760, 44432640, 10886400

Offset: 0

Alois P. Heinz, Sep 24 2018

			Triangle T(n,k) begins:
  1;
  0,   1;
  0,   3,     4;
  0,   6,    22,     18;
  0,  13,    96,    198,     120;
  0,  24,   330,   1272,    1800,     840;
  0,  48,  1146,   7518,   19152,   20640,    7920;
  0,  86,  3518,  36684,  148200,  274080,  234720,   75600;
  0, 160, 10946, 177438, 1080960, 3083640, 4462560, 3180240, 887040;
  ...

Alois P. Heinz, Rows n = 0..50, flattened
Eric Weisstein's World of Mathematics, Plane partition
Wikipedia, Plane partition

Columns k=0-1 give: A000007, A000219 (for n>0).
Row sums give A319601.
Main diagonal gives A053529.
Cf. A255970, A319730.

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A306100(n,k-i).

T(n,k) = k! * A319730(n,k).

A323450 Number of ways to fill a Young diagram with positive integers summing to n such that all rows and columns are weakly increasing.

Original entry on oeis.org

1, 1, 3, 6, 14, 26, 56, 103, 203, 374, 702, 1262, 2306, 4078, 7242, 12628, 21988, 37756, 64682, 109606, 185082, 309958, 516932, 856221, 1412461, 2316416, 3783552

Offset: 0

Gus Wiseman, Jan 16 2019

A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers.

			The a(4) = 14 generalized Young tableaux:
  4   1 3   2 2   1 1 2   1 1 1 1
.
  1   2   1 1   1 2   1 1   1 1 1
  3   2   2     1     1 1   1
.
  1   1 1
  1   1
  2   1
.
  1
  1
  1
  1
The a(5) = 26 generalized Young tableaux:
  5   1 4   2 3   1 1 3   1 2 2   1 1 1 2   1 1 1 1 1
.
  1   2   1 1   1 3   1 2   1 1   1 1 1   1 1 2   1 1 1   1 1 1 1
  4   3   3     1     2     1 2   2       1       1 1     1
.
  1   1   1 1   1 2   1 1   1 1 1
  1   2   1     1     1 1   1
  3   2   2     1     1     1
.
  1   1 1
  1   1
  1   1
  2   1
.
  1
  1
  1
  1
  1

nLab, Young Diagram.
The Unapologetic Mathematician weblog, Generalized Young Tableaux.

Cf. A000085, A000219, A003293, A053529, A114736, A138178, A296188, A299968.

Cf. A323436, A323437, A323438, A323439, A323451.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Sum[Length[Select[ptnplane[Times@@Prime/@y],And@@(LessEqual@@@Transpose[PadRight[#]/.(0->Infinity)])&]],{y,IntegerPartitions[n]}],{n,10}]

a(16)-a(26) from Seiichi Manyama, Aug 19 2020

A362827 Array read by antidiagonals: T(n,k) is the number of k-tuples of permutations of [n] that pairwise commute.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 6, 1, 1, 1, 8, 18, 24, 1, 1, 1, 16, 48, 120, 120, 1, 1, 1, 32, 126, 504, 840, 720, 1, 1, 1, 64, 336, 2016, 4680, 7920, 5040, 1, 1, 1, 128, 918, 7944, 24720, 66240, 75600, 40320, 1, 1, 1, 256, 2568, 31200, 130440, 516240, 856800, 887040, 362880, 1

Offset: 0

Andrew Howroyd, May 08 2023

Two permutations x,y on [n] commute if x*y = y*x.

			Array begins:
========================================================
n/k| 0    1     2      3       4        5          6 ...
---+----------------------------------------------------
0  | 1    1     1      1       1        1          1 ...
1  | 1    1     1      1       1        1          1 ...
2  | 1    2     4      8      16       32         64 ...
3  | 1    6    18     48     126      336        918 ...
4  | 1   24   120    504    2016     7944      31200 ...
5  | 1  120   840   4680   24720   130440     699840 ...
6  | 1  720  7920  66240  516240  3968640   30672720 ...
7  | 1 5040 75600 856800 9122400 97030080 1050336000 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals).
Tad White, Counting Free Abelian Actions, arXiv preprint arXiv:1304.2830 [math.CO], 2013.

Columns k=0..3 are A000012, A000142, A053529, A072169.
Main diagonal is A362828.
Cf. A362824, A362826.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
M(n,m=n)={my(v=vector(m+1), u=vector(n,n,n==1), f=vector(n,n,n!)); v[1]=vectorv(n+1,i,1); for(j=1, #v-1, my(t=EulerT(u)); v[j+1]=vectorv(n+1,i,i--;if(i,f[i]*t[i],1)); u=dirmul(u, vector(n, n, n^(j-1)))); Mat(v)}
{ my(A=M(7)); for(n=1, #A, print(A[n,])) }

T(n,k) = n!*A362826(n,k) for k > 0.

A234937 Triangle read by rows of coefficients of polynomials generated by the Han/Nekrasov-Okounkov formula.

Original entry on oeis.org

1, 1, -1, 4, -5, 1, 18, -29, 12, -1, 120, -218, 119, -22, 1, 840, -1814, 1285, -345, 35, -1, 7920, -18144, 14674, -5205, 805, -51, 1, 75600, -196356, 185080, -79219, 16450, -1624, 70, -1, 887040, -2427312, 2515036, -1258628, 324569, -43568, 2954, -92, 1

Offset: 0

William J. Keith, Jan 01 2014

Coefficients of the polynomials p_n(b) defined by Product_{k>0} (1-q^k)^(b-1) = Sum n! p_n(b) q^n.
Each row is length 1+n, starting from n=0, and consists of the coefficients of one of the p_n(b).
A210590 is an unsigned version using the form preferred by Nekrasov and Okounkov. This is the form for which Guo-Niu Han's reference below gives the hooklength formula:
p_n(b) = Sum_{lambda partitioning n} Product_{h_{ij} in lambda} (1-b/(h_{ij}^2)).
Coefficients reduced mod 5 are those of 2 times Pascal's triangle and an alternating sign. Other primes have slightly more complex reduction behavior. See second link.
Lehmer's conjecture on the tau function states that the evaluation at b=25 (A000594) is never 0.
The general diagonal and column are probably of combinatorial interest.

			The coefficient of q^3 in the indeterminate power is (1/6) (18-29b+12b^2-b^3).

Seiichi Manyama, Rows n = 0..100, flattened
G.-N. Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO], 2008.
W. J. Keith, Polynomial analogues of Ramanujan congruences for Han's hooklength formula, arXiv:1109.1236 [math.CO], 2011-2012; Acta Arith. 160 (2013), 303-315.

Row entries sum to 0.
A210590 is the unsigned version.
Starting from row 0: final entry of row n, (-1)^n (A033999).
From row 1: next-to-last entry of row n, (-1)^(n-1) * n(3n-1)/2 (signed version of A000326).
First entry of row n, n! * p(n) (A053529).
Second entry of row n, -1 * n! * (sum of reciprocals of all parts in partitions of n) (negatives of A057623).
(Sum of absolute values of row entries)/n!: A000712.
Evaluations at various powers of b, divided by n!, enumerate multipartitions or powers of the eta function. Some special cases that appear in the OEIS:
b=0: A000041, the partition numbers,
b=2: A010815, from Euler's Pentagonal Number Theorem,
b=-1: A000712, partitions into 2 colors,
b=-11: A005758, reciprocal of the square root of the tau function,
b=-23: A006922, reciprocal of the tau function,
b=13: A000735, square root of the tau function,
b=25: A000594, Ramanujan's tau function.

nn=10;
Clear[b]; PolyTable = Table[0, {n, 1, nn}];
PolyTable[[1]]=1-b;
For[n = 2, n <= nn, n++,
PolyTable[[n]] = Simplify[(((n - 1)!)*(b - 1))*(Sum[
       PolyTable[[n - m]]*(-1*DivisorSigma[1, m]/((n - m)!)), {m, 1,
        n - 1}] + (-1*DivisorSigma[1, n]))]];
LongTable = Table[Table[
   Which[k == 0, PartitionsP[n]*n!, k > 0,
    Coefficient[Expand[PolyTable[[n]]], b^k]], {k, 0, n}], {n, 1, nn}];
Flatten[PrependTo[LongTable,1]]

E.g.f.: Product_{k>0} (1-q^k)^(b-1).

Recurrence: With p_0(b) = 1, p_n(b) = (n-1)!*(b-1)*Sum_{m=1..n} -sigma(m)*p_{n-m}(b) / (n-m)!, sigma being the divisor function.

cp's OEIS Frontend

A210590 Triangle of numbers generated by the Nekrasov-Okounkov formula.

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A239841 Ordered pairs of permutation functions on n elements where f(g(g(x))) = g(g(f(x))).

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A306042 Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1 + x)^k).

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Maple

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A072169 Commuting permutations: number of ordered triples of permutations f, g, h in Symm(n) which all commute.

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A239836 Number of ordered pairs of permutation functions f,g on a size n set where f(g(g(x))) = g(f(f(x))).

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A305127 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^(sigma(k)/k), where sigma(k) is the sum of the divisors of k.

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A319600 Number T(n,k) of plane partitions of n into parts of exactly k sorts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A323450 Number of ways to fill a Young diagram with positive integers summing to n such that all rows and columns are weakly increasing.

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A362827 Array read by antidiagonals: T(n,k) is the number of k-tuples of permutations of [n] that pairwise commute.