A110141 -id:A110141 - cp's OEIS frontend

A110143 Row sums of triangle A110141.

1, 1, 4, 11, 43, 161, 901, 5579, 43206, 378360, 3742738, 40853520, 488029621, 6323154547, 88308425755, 1322120265238, 21122364398761, 358647945023885, 6449299885654827, 122436442904193940, 2447046870232798369, 51358050784584629338, 1129314001779283063606

Offset: 0

Paul D. Hanna, Jul 14 2005

nonn

Row n of triangle A110141 lists the denominators of unit fraction coefficients of the products of {c_k}, in ascending order by indices of {c_k}, in the coefficient of x^n in exp(Sum_{k>=1} c_k/k*x^k). There are A000041(n) terms in row n of triangle A110141.
Also, number of orbits of Sym(n)^2 where Sym_n acts by conjugation.  Compare the MathOverflow discussion, also Bogaerts-Dukes 2014, and A241584, A241585. - Peter J. Dukes, May 12 2014
Number of isomorphism classes of n-fold coverings of a connected graph with circuit rank 2 [Kwak and Lee]. - Álvar Ibeas, Mar 25 2015

P. A. MacMahon, The expansion of determinants and permanents in terms of symmetric functions, in Proc. ICM Toronto (ed. J. C. Fields), Toronto University Press, 1924, vol 1, 319-330.
J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. Appears to contain this sequence in Table 2. [Added by N. J. A. Sloane, Nov 12 2009]

Alois P. Heinz, Table of n, a(n) for n = 0..450
Mathieu Bogaerts and Peter Dukes, Semidefinite programming for permutation codes, Discrete Math. 326 (2014), 34--43. MR3188985.
Nicholas Dub, Enumeration of triangulations modulo symmetries and of rooted triangulations counted by their number of (d - 2)-simplices in dimension d ≥ 2, tel-03641958 [cs.OH], Université Paris-Nord - Paris XIII, 2021.
J. B. Geloun, S. Ramgoolam, Counting Tensor Model Observables and Branched Covers of the 2-Sphere, arXiv:1307.6490 [hep-th], 2013.
Joseph Ben Geloun, Sanjaye Ramgoolam, All-orders asymptotics of tensor model observables from symmetries of restricted partitions, arXiv:2106.01470 [hep-th], Jun 02 2021.
J. H. Kwak and J. Lee, Isomorphism classes of graph bundles. Can. J. Math., 42(4), 1990, pp. 747-761.
MathOverflow, A general formula for the number of conjugacy classes of S_n×S_n acted on by S_n [From Peter Dukes, May 12 2014]
Igor Pak, Greta Panova, Damir Yeliussizov, On the largest Kronecker and Littlewood-Richardson coefficients, arXiv:1804.04693 [math.CO], 2018.

Cf. A110141, A000041, A057005, A096161, A241584, A241585, A279819.

Third column of A160449.

# Using a function from Alois P. Heinz in A279038:
b:= proc(n, i) option remember; `if`(n=0, [1],
      `if`(i<1, [], [seq(map(x-> x*i^j*j!,
      b(n-i*j, i-1))[], j=0..n/i)]))
    end:
seq(add(i, i=b(n$2)), n=0..22); # Peter Luschny, Dec 19 2016

Table[Total[Apply[Times, Tally[#]/.{a_Integer,b_}->a^b b!]& /@ IntegerPartitions[n]],{n,0,21}] (* Wouter Meeussen, Oct 17 2014 *)
b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i < 1, {}, Flatten[ Table[ Map[ #*i^j*j!&, b[n-i*j, i-1]], {j, 0, n/i}]]]]; Table[Sum[i, {i, b[n, n]}], {n, 0, 22}] (* Jean-François Alcover, Jul 10 2017, after Alois P. Heinz *)
nmax = 25; CoefficientList[Series[Product[Sum[k!*j^k*x^(j*k), {k, 0, nmax/j}], {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 08 2019 *)
m = 30; CoefficientList[Series[Product[-Gamma[0, -1/(x^j*j)] * Exp[-1/(x^j*j)], {j, 1, m}] / (x^(m*(m + 1)/2)*m!), {x, 0, m}], x] (* Vaclav Kotesovec, Dec 07 2020 *)

def A110143(n):
    return sum(p.aut() for p in Partitions(n))
[A110143(n) for n in range(9)]
# Álvar Ibeas, Mar 26 2015

G.f.: B(x)*B(2*x^2)*B(3*x^3)*..., where B(x) is g.f. of A000142. - Vladeta Jovovic, Feb 18 2007

a(n) ~ n! * (1 + 2/n^2 + 5/n^3 + 23/n^4 + 106/n^5 + 537/n^6 + 3143/n^7 + 20485/n^8 + 143747/n^9 + 1078660/n^10), for coefficients see A279819. - Vaclav Kotesovec, Mar 16 2015

A110142 Limit of rows of triangle A110141 after dividing respectively by a list of factorials, with (n-j-1)! repeated A002865(j+1) times in the list as j=1..n-1.

Original entry on oeis.org

1, 2, 3, 8, 4, 6, 5, 48, 8, 18, 6, 24, 10, 12, 7, 384, 32, 36, 12, 15, 32, 8, 144, 40, 24, 14, 162, 18, 20, 9, 3840, 192, 144, 48, 30, 64, 16, 72, 21, 24, 50, 10, 1152, 240, 96, 56, 324, 36, 40, 18, 90, 96, 24, 28, 30, 11, 46080, 1536, 864, 288, 120, 256, 64, 144, 42, 48, 100

Offset: 1

Paul D. Hanna, Jul 13 2005

nonn

Row n of triangle A110141 lists the denominators of unit fraction coefficients of the products of {c_k}, in ascending order by indices of {c_k}, in the coefficient of x^n in exp(Sum_{k>=1} c_k/k*x^k). A002865 equals the first differences of the partition numbers. A110144 lists terms at positions p(n)+1.

			Row 6 of A110141 is: {720,48,18,16,8,6,5,48,8,18,6};
divided respectively by: {6!,4!,3!,2!,2!,1!,1!,0!,0!,0!,0!}
with {4!,3!,2!,1!,0!} each occurring {1,1,2,2,4} times after 6!,
yields the initial A000041(6)=11 terms: {1,2,3,8,4,6,5,48,8,18,6}.
Sum of reciprocal terms at positions p(5)+1 through p(6) =
1/48 + 1/8 + 1/18 + 1/6 = 1-1+1/2!-1/3!+1/4!-1/5!+1/6!.
Other patterns emerge when the terms are read by groups
of terms in positions p(n-1)+1 through p(n):
1;
2;
3;
8,4;
6, 5;
48,8, 18,6;
24,10, 12,7;
384,32,36,12, 15,32,8;
144,40,24,14, 162,18,20,9;
3840,192,144,48,30,64,16, 72,21,24,50,10;
1152,240,96,56,324,36,40,18, 90,96,24,28,30,11;
46080,1536,864,288,120,256,64,144,42,48,100,20, 1944,108,60,27,384,32,35,72,12;
11520,1920,576,336,1296,144,160,72,180,192,48,56,60,22, 648,126,72,150,30,160,36,40,42,13; ...

Cf. A110141, A110144, A000041, A002865.

a(p(n)) = n where p(n) = A000041(n) (partition numbers) for n>=1. Sum_{k=p(n-1)+1..p(n)} 1/a(k) = Sum_{k=0..n} (-1)^k/k!, for n>1.

A102189 Array of multinomial numbers (row reversed order of table A036039).

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 3, 8, 6, 1, 10, 15, 20, 20, 30, 24, 1, 15, 45, 40, 15, 120, 90, 40, 90, 144, 120, 1, 21, 105, 70, 105, 420, 210, 210, 280, 630, 504, 420, 504, 840, 720, 1, 28, 210, 112, 420, 1120, 420, 105, 1680, 1120, 2520, 1344, 1120, 1260, 3360, 4032, 3360

Offset: 1

Wolfdieter Lang, Feb 15 2005

See Abramowitz and Stegun, Handbook, p. 831, column labeled "M_2", read backwards.
The sequence of row lengths is [1,2,3,5,7,11,15,...] = A000041(n), n>=1 (partition numbers).
Row n of this array gives the coefficients of the cycle index polynomial n!*Z(S_n) for the symmetric group S_n. For instance, Z(S_4)= (x[1]^4 + 6*x[1]^2*x[2] + 3*x[2]^2 + 8*x[1]*x[3] + 6*x[4])/4!. The partitions of 4 appear here in the reversed Abramowitz-Stegun order.
See the W. Lang link "Solution of Newton's Identities" and the Note added Jun 06 2007 in the link "More rows and S_n cycle index polynomials" for the appearance of the signed array. - Wolfdieter Lang, Aug 01 2013
Multiplying the values of row n by the corresponding values in row n of A110141, one obtains n!. - Jaimal Ichharam, Aug 06 2015

			Triangle begins:
  [1];
  [1,1];
  [1,3,2];
  [1,6,3,8,6];
  [1,10,15,20,20,30,24];
  ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 831-2.
Wolfdieter Lang, More rows and S_n cycle index polynomials.
Wolfdieter Lang, Solution of Newton's Identities.
Andrei Vieru, Analytic renormalization of multiple zeta functions. Geometry and combinatorics of generalized Euler reflection formula for MZV, arXiv preprint arXiv:1601.04703 [math.NT], 2016.

Cf. A000041, A036039, A110141.

aspartitions[n_] := Reverse /@ Sort[Sort /@ IntegerPartitions[n]]; ascycleclasses[n_Integer] := n!/(Times @@ #)& /@ ((#! Range[n]^#)& /@ Function[par, Count[par, #]& /@ Range[n]] /@ aspartitions[n]); row[n_] := ascycleclasses[n] // Reverse; Table[row[n], {n, 1, 8}] // Flatten (* Jean-François Alcover, Feb 04 2014, after A036039 and Wouter Meeussen *)

A110144 Terms of A110142 at positions p(n)+1, where p(n) = A000041(n) is the number of partitions of n; a(n) = A110142(p(n)+1) for n>=1, with a(0) = 1.

Original entry on oeis.org

1, 2, 3, 8, 6, 48, 24, 384, 144, 3840, 1152, 46080, 11520, 645120, 138240, 10321920, 1935360, 185794560, 30965760, 3715891200, 557383680, 81749606400, 11147673600, 1961990553600, 245248819200, 51011754393600, 5885971660800

Offset: 0

Paul D. Hanna, Jul 14 2005

nonn

Cf. A110142, A110141, A000041, A000165 (double factorials).

a(n)=if(n==0,1,if(n%2==1,2^(n\2+1)*(n\2+1)!,3*2^((n-1)\2)*((n-1)\2)!))

a(2*n+1) = 2^n*n!, a(2*n+2) = 3*2^n*n! for n>0, with a(0) = 1.

A341105 T(n, k) is the Cauchy coefficient of the k-th partition of n, where the partitions are enumerated in standard order. T(n, k) for n >= 0 and 1 <= k <= A000041(n).

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 6, 4, 3, 8, 4, 24, 5, 4, 6, 6, 8, 12, 120, 6, 5, 8, 8, 18, 6, 18, 48, 16, 48, 720, 7, 6, 10, 10, 12, 8, 24, 18, 24, 12, 72, 48, 48, 240, 5040, 8, 7, 12, 12, 15, 10, 30, 32, 12, 32, 16, 96, 36, 36, 24, 36, 360, 384, 96, 192, 1440, 40320

Offset: 0

Peter Luschny, Feb 25 2021

By the 'standard order' of partitions we understand the graded reverse lexicographic ordering A080577.

We call the coefficients the 'Cauchy coefficients' because they were used by Cauchy in his proof of the number of permutations on [n] with cycle structure p.

			Triangle begins:
[0] [1]
[1] [1]
[2] [2, 2]
[3] [3, 2, 6]
[4] [4, 3, 8, 4, 24]
[5] [5, 4, 6, 6, 8, 12, 120]
[6] [6, 5, 8, 8, 18, 6, 18, 48, 16, 48, 720]
[7] [7, 6, 10, 10, 12, 8, 24, 18, 24, 12, 72, 48, 48, 240, 5040]
.
For instance, the 40th partition of n = 12 is [5, 2, 2, 2, 1], and has the frequency vector [1, 3, 0, 0, 1]. Thus T(12, 40) = (1!*1^1)*(3!*2^3)*(1!*5^1) = 240. To compute this value with the Sage program below invoke list(A341105row(12))[40].

The row terms are a permutation of the row terms of A110141.

Cf. A000041, A080577.

def PartitionsFreq(n): # returns a generator object
    return ([sum((1 if v == m else 0) for j, v in enumerate(p)) for m in (1..n)]
            for p in Partitions(n))
def A341105row(n): # returns a generator object
    return (product(factorial(p[i])*(i+1)^p[i] for i in range(n))
            for p in PartitionsFreq(n))
for n in range(9): print(list(A341105row(n)))

Let p be the k-th partition of n with frequency vector f. Then T(n, k) = Product_{i=1..n} f[i]! * i^f[i].

cp's OEIS Frontend

A110143 Row sums of triangle A110141.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A110142 Limit of rows of triangle A110141 after dividing respectively by a list of factorials, with (n-j-1)! repeated A002865(j+1) times in the list as j=1..n-1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A102189 Array of multinomial numbers (row reversed order of table A036039).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A110144 Terms of A110142 at positions p(n)+1, where p(n) = A000041(n) is the number of partitions of n; a(n) = A110142(p(n)+1) for n>=1, with a(0) = 1.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A341105 T(n, k) is the Cauchy coefficient of the k-th partition of n, where the partitions are enumerated in standard order. T(n, k) for n >= 0 and 1 <= k <= A000041(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

SageMath

Formula