Richard Stanley - cp's OEIS frontend

A384800 a(n) = A384727(A368538(n)).

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 6, 1, 1, 7, 1, 1, 5, 1, 6, 1, 1, 2, 4, 1, 1, 23, 1, 13, 2

Offset: 1

Richard Stanley, Jun 10 2025

Let 1=b_1A368538). Then a(n) is the number of such groups (up to isomorphism) of order b_n.

			Of the groups of order at most six, the 1-element group, 2-element group, and the symmetric group S_3 of order six are the only ones with the same number of elements as subgroups. Hence a(1) = a(2) = a(3) = 1.

Dave Benson, Congruence mod four of the number of subgroups of a finite 2-group, discussion in MathOverflow, 2025 Jun 11.

Cf. A368538, A384727.

// Output of A368538(n) and a(n)
limit := 104;
for i in [1 .. limit] do
  j := 0;
  for G in SmallGroups(i) do
    if #AllSubgroups(G) eq i then j +:= 1; end if;
  end for;
  if j gt 0 then i, j; end if;
end for; // Hugo Pfoertner, Jun 10 2025

A368538:= [1, 2, 6, 8, 28, 36, 40, 48, 54, 72, 96, 100, 104, 128, 132, 144, 160, 176, 180, 192, 216, 240, 252, 260, 288, 324, 336, 368, 384, 416, 456, 480, 496]:
seq(nops(select(g -> nops(convert(SubgroupLattice(g),list))=k, [seq(SmallGroup(k,i),i=1..NumGroups(k))])),k=A368538); # Robert Israel, Jun 10 2025

a(25)-a(32) from Richard Stanley, Jun 11 2025 using results by Dave Benson in MathOverflow discussion.

A384727 Number of groups of order n (up to isomorphism) with exactly n subgroups.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1

Offset: 1

Richard Stanley, Jun 08 2025

nonn

See A384800 for more information.

			The symmetric group S_3 has six elements and six subgroups. The other group of order six has four subgroups, so a(6)=1.

Richard Stanley, Table of n, a(n) for n = 1..192
Richard Stanley, What finite groups have as many elements as subgroups? Question in Mathoverflow, answered by Dave Benson and others, Jun 07 2025.

Cf. A368538, A384800.

A380394 a(n) = number of possible pairs of descent sets of a permutation of 1,2,...,n and its inverse.

Original entry on oeis.org

1, 1, 2, 6, 22, 94, 426, 1938, 8724, 38724, 169438, 731390, 3119052, 13162228

Offset: 0

Richard Stanley, Jan 23 2025

			For n=3 there are the six pairs (empty,empty), (1,1), (1,2), (2,1), (2,2), (12,12).

Richard Stanley, Descent sets and inverse descent sets of a permutation, MathOverflow, 2025.

def a380394(n): return len({ (tuple(p.descents()), tuple(p.idescents())) for p in Permutations(n) }) # Max Alekseyev, Jan 24 2025

a(0)=1 prepended by Alois P. Heinz, Jan 25 2025

a(11)-a(13) from Max Alekseyev, Jan 28 2025

A362153 Number of skew shapes in a 3 X n rectangle with no empty rows or columns.

Original entry on oeis.org

1, 8, 29, 73, 151, 276, 463, 729, 1093, 1576, 2201, 2993, 3979, 5188, 6651, 8401, 10473, 12904, 15733, 19001, 22751, 27028, 31879, 37353, 43501, 50376, 58033, 66529, 75923, 86276, 97651, 110113, 123729, 138568, 154701, 172201, 191143, 211604, 233663, 257401, 282901, 310248, 339529

Offset: 1

Richard Stanley, Apr 09 2023

			The a(2) = 8 shapes are 211, 221, 222, 221/1, 222/1, 222/11, 221/11, 211/1.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

a := proc(n) m := n*(n + 4); m*(m - 5) / 12 + 1 end:
seq(a(n), n = 1..43); # Peter Luschny, Apr 10 2023

a(n) = (n^4 + 8*n^3 + 11*n^2 - 20*n + 12)/12.

a(n) = [x^n] (-3*x^4 + 11*x^3 - 13*x^2 + 4*x - 1)/(x - 1)^5. - Peter Luschny, Apr 10 2023

A360808 Number of double cosets of the Sylow 2-subgroup of the symmetric group S_n.

Original entry on oeis.org

1, 1, 2, 2, 4, 8, 35, 16, 51, 145, 1112, 1145, 10929, 41400, 542785, 40384, 583169, 2781808, 48558706, 65461347, 1277941540, 7370563251, 159694747220, 63387056365, 1500631724572, 10152855622657

Offset: 1

Richard Stanley, Feb 21 2023

Let S denote a Sylow 2-subgroup of the symmetric group S_n. Then a(n) is the number of double cosets SwS. [Corrected by Benjamin Sambale, Mar 08 2025]

Persi Diaconis, Eugenio Giannelli, Robert M. Guralnick, Stacey Law, Gabriel Navarro, Benjamin Sambale, and Hunter Spink, On the number and sizes of double cosets of Sylow subgroups of the symmetric group, arXiv:2504.01149 [math.GR], 2025. See pp. 5, 15.

Define a symmetric function T_k recursively by T_0 = p_1 (power sum), and T_k is the plethysm h_2[T_{k-1}] for k>0. If n has the binary expansion 2^{a_0} + 2^{a_1} + ..., then set $U_n = T_{a_0}T_{a_1}... Then a_n = (usual scalar product on symmetric functions).

a(2) and a(5) corrected by Benjamin Sambale, Mar 08 2025

A352457 Codimension of Lyndon symmetric functions of degree n.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 3, 4, 4, 4, 7, 10, 12, 15, 24, 27, 31, 40, 50, 59, 78

Offset: 1

Richard Stanley, Mar 16 2022

For each partition p of n, there is a "Lyndon symmetric function" L_p which is homogeneous of degree n. The Q-vector space V_n spanned by all Lyndon symmetric functions of degree n is a subspace of the space Lambda^n of all homogeneous symmetric functions over Q of degree n, which has dimension p(n), the number of partitions of n (A000041). Then a(n) is the codimension of V_n in Lambda^n.

R. Stanley, Enumerative Combinatorics, volume 2, Exercise 7.89.

Cf. A000041.

More terms from Ira M. Gessel, Mar 20 2025

A337083 Number of spanning trees of the 1-skeleton of the (n-1)-dimensional permutohedron.

Original entry on oeis.org

1, 1, 6, 101154816, 6187732257761496793412385090375984958331031826464768000000000

Offset: 1

Richard Stanley, Aug 14 2020

nonn

We have the factorizations:
a(4) = 2^15 * 3^2 * 7^3.
a(5) = 2^59 * 3^15 * 5^9 * 7^5 * 11^6 * 23^5 * 29^4 * 41^4.
a(6) = 2^215 * 3^178 * 5^47 * 7^15 * 11^39 * 13^10 * 19^16 * 23^15 * 29^16 * 41^16 * 61^5 * 67^9 * 71^5 * 1931^16 * 3253^9.

			For n=3 the permutohedron is a hexagon, which has six spanning trees.

Richard Stanley, Table of n, a(n) for n = 1..6
Eric Weisstein's World of Mathematics, Bruhat Graph
Wikipedia, Permutohedron

Cf. A006237.

import sympy,itertools
def A337083(n):
  p=tuple(itertools.permutations(range(n)))
  m=len(p)
  q={p[i]:i for i in range(m)}
  Q=sympy.diag(*[n-1]*m)
  for i in range(m):
    for k in range(n-1):
      Q[i,q[p[i][:k]+tuple(reversed(p[i][k:k+2]))+p[i][k+2:]]]=-1
  return Q[:m-1,:m-1].det() # Pontus von Brömssen, Jan 18 2021

a(1) prepended by Pontus von Brömssen, Jan 19 2021

A336525 Total number of linear extensions of all n-element posets.

Original entry on oeis.org

1, 1, 3, 14, 96, 895, 11751, 214708, 5594463

Offset: 0

Richard Stanley, Jul 27 2020

Sum of e(P) over all nonisomorphic n-element posets, where e(P) is the number of linear extensions of P.

			There is one 3-element poset with 6 linear extensions, one with 3, two with 2, and one with 1, for a total of 14.

Cf. A006251, A080687, A248091, A301871, A336229, A006455.

A336229 Nonisomorphic n-element posets whose order polynomial factors into linear factors over the rationals.

Original entry on oeis.org

1, 2, 5, 14, 31, 111, 257, 624

Offset: 1

Richard Stanley, Jul 12 2020

			For n <= 3 all posets are counted.

Sam Hopkins, Order polynomial product formulas and poset dynamics, arXiv 2006.01568 [math.CO], 2020.
MathOverflow, Unified framework for posets with order polynomial product formulas, 2019.

A333331 Number of integer points in the convex hull in R^n of parking functions of length n.

Original entry on oeis.org

1, 3, 17, 144, 1623, 22804, 383415, 7501422

Offset: 1

Richard Stanley, Mar 15 2020

It is observed by Gus Wiseman in A368596 and A368730 that this sequence appears to be the complement of those sequences. If this is the case, then a(n) is the number of labeled graphs with loops allowed in which each connected component has an equal number of vertices and edges and the conjectured formula holds. Terms for n >= 9 are expected to be 167341283, 4191140394, 116425416531, ... - Andrew Howroyd, Jan 10 2024
From Gus Wiseman, Mar 22 2024: (Start)
An equivalent conjecture is that a(n) is the number of loop-graphs with n vertices and n edges such that it is possible to choose a different vertex from each edge. I call these graphs choosable. For example, the a(3) = 17 choosable loop-graphs are the following (loops shown as singletons):
  {{1},{2},{3}}  {{1},{2},{1,3}}  {{1},{1,2},{1,3}}  {{1,2},{1,3},{2,3}}
                 {{1},{2},{2,3}}  {{1},{1,2},{2,3}}
                 {{1},{3},{1,2}}  {{1},{1,3},{2,3}}
                 {{1},{3},{2,3}}  {{2},{1,2},{1,3}}
                 {{2},{3},{1,2}}  {{2},{1,2},{2,3}}
                 {{2},{3},{1,3}}  {{2},{1,3},{2,3}}
                                  {{3},{1,2},{1,3}}
                                  {{3},{1,2},{2,3}}
                                  {{3},{1,3},{2,3}}
(End)

			For n=2 the parking functions are (1,1), (1,2), (2,1). They are the only integer points in their convex hull. For n=3, in addition to the 16 parking functions, there is the additional point (2,2,2).

R. P. Stanley (Proposer), Problem 12191, Amer. Math. Monthly, 127:6 (2020), 563.

Thomas Selig, The stochastic sandpile model on complete graphs, arXiv:2209.07301 [math.PR], 2022.

All of the following relative references pertain to the conjecture:
The case of unique choice A000272.
The version without the choice condition is A014068, covering A368597.
The case of just pairs A137916.
For any number of edges of any positive size we have A367902.
The complement A368596, covering A368730.
Allowing edges of any positive size gives A368601, complement A368600.
Counting by singletons gives A368924.
For any number of edges we have A368927, complement A369141.
The connected case is A368951.
The unlabeled version is A368984, complement A368835.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts graphs, unlabeled A000088.
A058891 counts set-systems (without singletons A016031), unlabeled A000612.
Cf. A000169, A000666, A057500, A062740, A129271, A133686, A367863, A367903, A369146, A369199.

Conjectured e.g.f.: exp(-log(1-T(x))/2 + T(x)/2 - T(x)^2/4) where T(x) = -LambertW(-x) is the e.g.f. of A000169. - Andrew Howroyd, Jan 10 2024

cp's OEIS Frontend

User: Richard Stanley

A384800 a(n) = A384727(A368538(n)).

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A384727 Number of groups of order n (up to isomorphism) with exactly n subgroups.

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A380394 a(n) = number of possible pairs of descent sets of a permutation of 1,2,...,n and its inverse.

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Sage

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A362153 Number of skew shapes in a 3 X n rectangle with no empty rows or columns.

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Maple

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A360808 Number of double cosets of the Sylow 2-subgroup of the symmetric group S_n.

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Formula

Extensions

A352457 Codimension of Lyndon symmetric functions of degree n.

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A337083 Number of spanning trees of the 1-skeleton of the (n-1)-dimensional permutohedron.

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Python

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A336525 Total number of linear extensions of all n-element posets.

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A336229 Nonisomorphic n-element posets whose order polynomial factors into linear factors over the rationals.

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