A002873 -id:A002873 - cp's OEIS frontend

A002872 Number of partitions of {1..2n} that are invariant under a permutation consisting of n 2-cycles.

1, 2, 7, 31, 164, 999, 6841, 51790, 428131, 3827967, 36738144, 376118747, 4086419601, 46910207114, 566845074703, 7186474088735, 95318816501420, 1319330556537631, 19013488408858761, 284724852032757686, 4422344774431494155, 71125541977466879231

Offset: 0

N. J. A. Sloane, Simon Plouffe

Previous name was: Sorting numbers.
a(n) = number of symmetric partitions of the set {-n,...,-1,1,...,n}. A partition of {-n,...,-1,1,...,n} into nonempty subsets X_1,...,X_k is 'symmetric' if for each i, -X_i=X_j for some j. a(n) = S_B(n,1)+...+S_B(n,n) where S_B(n,k) is as in A085483. a(n) is the n-th Bell number of 'type B'. - James East, Aug 18 2003
Column 2 of A162663. - Franklin T. Adams-Watters, Jul 09 2009
a(n) is equal to the sum of all expressions of the form p(1^n)[st(lambda)] for partitions lambda of order less than or equal to n, where p(1^n)[st(lambda)] denotes the coefficient of the irreducible character basis element indexed by the partition lambda in the expansion of the power sum basis element indexed by the partition (1^n). - John M. Campbell, Sep 16 2017
Number of achiral color patterns in a row or loop of length 2n. Two color patterns are equivalent if the colors are permuted. - Robert A. Russell, Apr 24 2018
Stirling transform of A005425 per Knuth reference. - Robert A. Russell, Apr 28 2018

			For a(2)=7, the row patterns are AAAA, AABB, ABAB, ABBA, ABBC, ABCA, and ABCD.  The loop patterns are AAAA, AAAB, AABB, AABC, ABAB, ABAC, and ABCD. - _Robert A. Russell_, Apr 24 2018

D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.5 (p. 765). - Robert A. Russell, Apr 28 2018
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..513 (first 101 terms from T. D. Noe)
Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See pp. 12, 18, 29.
T. Halverson and M. Reeks, Gelfand Models for Diagram Algebras, arXiv preprint arXiv:1302.6150 [math.RT], 2013.
Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.
Victor Meally, Comparison of several sequences given in Motzkin's paper "Sorting numbers for cylinders...", letter to N. J. A. Sloane, N. D.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
J. Pasukonis and S. Ramgoolam, From counting to construction for BPS states in N=4 SYM, arXiv:1010.1683 [hep-th], 2010, (E.2).
J. Pasukonis and S. Ramgoolam, From counting to construction for BPS states in N=4 SYM, J. High En. Phys. 2011 (2) (2011), (E.2).
OEIS Wiki, Sorting numbers
R. Orellana and M. Zabrocki, Symmetric group characters as symmetric functions, arXiv preprint arXiv:1605.06672 [math.CO], 2016-2017.
J. Quaintance, Letter representations of rectangular m x n x p proper arrays, arXiv:math/0412244 [math.CO], 2004-2006.
Index entries for sequences related to sorting

Cf. A002873, A002874, A080107, A085483.
u[n,j] is A162663.
Row sums of A293181.
Column k=2 of A306024.
Cf. A005425.

a:= proc(n) option remember; `if`(n=0, 1, add((1+
      2^(j-1))*binomial(n-1, j-1)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 29 2015

u[0,j_]:=1;u[k_,j_]:=u[k,j]=Sum[Binomial[k-1,i-1]Plus@@(u[k-i,j]#^(i-1)&/@Divisors[j]),{i,k}]; Table[u[n,2],{n,0,12}] (* Wouter Meeussen, Dec 06 2008 *)
mx = 16; p = 2; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
Aeven[m_, k_] := Aeven[m, k] = If[m>0, k Aeven[m-1, k] + Aeven[m-1, k-1]
  + Aeven[m-1, k-2], Boole[m==0 && k==0]]
Table[Sum[Aeven[m, k], {k, 0, 2m}], {m, 0, 30}] (* Robert A. Russell, Apr 24 2018 *)
x[n_] := x[n] = If[n<2, n+1, 2x[n-1] + (n-1)x[n-2]]; (* A005425 *)
Table[Sum[StirlingS2[n, k] x[k], {k, 0, n}], {n, 0, 20}] (* Robert A. Russell, Apr 28 2018, from Knuth reference *)
Table[Sum[Binomial[n,k] * 2^k * BellB[k, 1/2] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 29 2022 *)

E.g.f.: e^( (e^(2x) - 3)/2 + e^x ).
a(n) = A080107(2n) for all n. - Jörgen Backelin, Jan 13 2016
From Robert A. Russell, Apr 24 2018: (Start)
Aeven(n,k) = [n>0]*(k*Aeven(n-1,k)+Aeven(n-1,k-1)+Aeven(n-1,k-2))
  + [n==0]*[k==0]
a(n) = Sum_{k=0..2n} Aeven(n,k). (End)
a(n) = Sum_{k=0..n} Stirling2(n, k)*A005425(k). (from Knuth reference) - Robert A. Russell, Apr 28 2018
a(n) ~ exp(exp(2*r)/2 + exp(r) - 3/2 - n) * (n/r)^(n + 1/2) / sqrt((1 + 2*r)*exp(2*r) + (1 + r)*exp(r)), where r = LambertW(2*n)/2 - 1/(1 + 2/LambertW(2*n) + n^(1/2) * (1 + LambertW(2*n)) * (2/LambertW(2*n))^(3/2)). - Vaclav Kotesovec, Jul 03 2022
a(n) ~ (2*n/LambertW(2*n))^n * exp(n/LambertW(2*n) + (2*n/LambertW(2*n))^(1/2) - n - 7/4) / sqrt(1 + LambertW(2*n)). - Vaclav Kotesovec, Jul 10 2022

Edited by Franklin T. Adams-Watters, Jul 09 2009

A293181 Irregular triangle read by rows: T(n,k) is the number of k-partitions of {1..2n} that are invariant under a permutation consisting of n 2-cycles (1 <= k <= 2n).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 1, 7, 10, 9, 3, 1, 1, 15, 38, 53, 34, 18, 4, 1, 1, 31, 130, 265, 261, 195, 80, 30, 5, 1, 1, 63, 422, 1221, 1700, 1696, 1016, 515, 155, 45, 6, 1, 1, 127, 1330, 5369, 10143, 13097, 10508, 6832, 2926, 1120, 266, 63, 7, 1

Offset: 1

Andrew Howroyd, Oct 01 2017

See A002872 for detailed description.
T(m,k) is the number of achiral color patterns in a row or loop of length 2m using exactly k different colors. Two color patterns are equivalent if we permute the colors. - Robert A. Russell, Apr 24 2018
T(n,k) = coefficient of x^k for A(2,n)(x) in Gilbert and Riordan's article. - Robert A. Russell, Jun 14 2018

			Triangle begins:
  1,   1;
  1,   3,    2,    1;
  1,   7,   10,    9,     3,     1;
  1,  15,   38,   53,    34,    18,     4,    1;
  1,  31,  130,  265,   261,   195,    80,   30,    5,    1;
  1,  63,  422, 1221,  1700,  1696,  1016,  515,  155,   45,   6,  1;
  1, 127, 1330, 5369, 10143, 13097, 10508, 6832, 2926, 1120, 266, 63, 7, 1;
  ...
For T(2,2)=3, the row patterns are AABB, ABAB, and ABBA.  The loop patterns are AAAB, AABB, and ABAB. - _Robert A. Russell_, Apr 24 2018

Alois P. Heinz, Rows n = 1..100, flattened (first 30 rows from Andrew Howroyd)
Ira Gessel, What is the number of achiral color patterns for a row of n colors containing k different colors?, mathoverflow, Jan 30 2018.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Row sums are A002872.
Maximum row values are A002873.
Number of achiral color patterns of length odd n in A140735.
Column k=3 gives A056182.

(* Ach[n, k] is the number of achiral color patterns for a row or loop of n
  colors containing k different colors *)
Ach[n_, k_] := Ach[n, k] = Which[0==k, Boole[0==n], 1==k, Boole[n>0],
  OddQ[n], Sum[Binomial[(n-1)/2, i] Ach[n-1-2i, k-1], {i, 0, (n-1)/2}],
  True, Sum[Binomial[n/2-1, i] (Ach[n-2-2i, k-1]
  + 2^i Ach[n-2-2i, k-2]), {i, 0, n/2-1}]]
Table[Ach[n, k], {n, 2, 14, 2}, {k, 1, n}] // Flatten
(* Robert A. Russell, Feb 06 2018 *)
Table[Drop[MatrixPower[Table[Switch[j-i, 0, i-1, 1, 1, 2, 1, _, 0],
  {i, 1, 2n+1}, {j, 1, 2n+1}], n][[1]], 1], {n, 1, 10}] // Flatten
(* Robert A. Russell, Apr 14 2018 *)
Aeven[m_, k_] := Aeven[m, k] = If[m>0, k Aeven[m-1, k] + Aeven[m-1, k-1]
  + Aeven[m-1, k-2], Boole[m == 0 && k == 0]]
Table[Aeven[m, k], {m, 1, 10}, {k, 1, 2m}] // Flatten (* Robert A. Russell, Apr 24 2018 *)

\\ see A056391 for Polya enumeration functions
T(n,k) = 2*NonequivalentStructsExactly(CylinderPerms(2,n),k) - stirling(2*n,k,2);

seq(n)={Vec(serlaplace(exp(y*(exp(x + O(x*x^n))-1)+(1/2)*y^2*(exp(2*x + O(x*x^n))-1))) - 1)}
{my(T=seq(10)); for(n=1, #T, for(k=1, 2*n, print1(polcoeff(T[n], k), ", ")); print)} \\ Andrew Howroyd, Jan 31 2018

T(n,k) = coefficient of t^k x^n/n! in exp(t*(exp(x)-1)+(1/2)*t^2*(exp(2*x)-1)). - Ira M. Gessel, Jan 30 2018
T(m,k) = [m>0]*(k*T(m-1,k)+T(m-1,k-1)+T(m-1,k-2)) + [m==0]*[k==0]. - Robert A. Russell, Apr 24 2018
Conjecture: T(n,k) = R(n,k)-R(n,k-1), with R(n,k) = Sum_{m=0..k} m^n*A000085(m)*A038205(k-m)/(m!*(k-m)!). - Mikhail Kurkov, Jun 26 2018

A002875 Sorting numbers (see Motzkin article for details).

Original entry on oeis.org

1, 2, 4, 24, 128, 880, 7440

Offset: 0

N. J. A. Sloane

How is the sequence defined (see the links in A000262)? Also more terms would be welcome.

Based on the Motzkin article, where this sequence appears in the last row of the table on p. 173, one would expect that this sequence is the same as A294202. However, they seem to be unrelated. So the true definition of this sequence is a mystery. - Andrew Howroyd and Andrey Zabolotskiy, Oct 25 2017

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Victor Meally, Comparison of several sequences given in Motzkin's paper "Sorting numbers for cylinders...", letter to N. J. A. Sloane, N. D.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers
Index entries for sequences related to sorting

Cf. A000262, A001861, A002872, A002873, A002874, A036073, A294201, A294202.

A036074 Expansion of e.g.f. exp((exp(p*x) - p - 1)/p + exp(x)) for p=4.

Original entry on oeis.org

1, 2, 9, 55, 412, 3619, 36333, 408888, 5080907, 68914023, 1011165446, 15935379409, 268125052373, 4792458452162, 90605469012877, 1805135197261131, 37775862401203916, 827992670793489263

Offset: 0

N. J. A. Sloane

nonn

T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
T. S. Motzkin, Sorting numbers ...: for a link to an annotated scanned version of this paper see A000262.

Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
Index entries for sequences related to sorting

Cf. A001861, A002872, A002873, A002874, A002875, A036076, ...

mx = 16; p = 4; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
Table[Sum[Binomial[n,k] * 4^k * BellB[k, 1/4] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 29 2022 *)

a(n):=sum(sum(binomial(m,i)*sum(binomial(i,j)*(1/4)^j*(3*j+i)^n,j,0,i)*(-5/4)^(m-i),i,0,m)/m!,m,1,n); /* Vladimir Kruchinin, Sep 14 2010 */

a(n) = sum(sum(binomial(m,i)*sum(binomial(i,j)*(1/4)^j*(3*j+i)^n,j,0,i)*(-5/4)^(m-i),i,0,m)/m!,m,1,n), n > 0. - Vladimir Kruchinin, Sep 14 2010
a(n) ~ exp(exp(p*r)/p + exp(r) - 1 - 1/p - n) * (n/r)^(n + 1/2) / sqrt((1 + p*r)*exp(p*r) + (1 + r)*exp(r)), where r = LambertW(p*n)/p - 1/(1 + p/LambertW(p*n) + n^(1 - 1/p) * (1 + LambertW(p*n)) * (p/LambertW(p*n))^(2 - 1/p)) for p=4. - Vaclav Kotesovec, Jul 03 2022
a(n) ~ (4*n/LambertW(4*n))^n * exp(n/LambertW(4*n) + (4*n/LambertW(4*n))^(1/4) - n - 5/4) / sqrt(1 + LambertW(4*n)). - Vaclav Kotesovec, Jul 10 2022

Edited by N. J. A. Sloane, Jul 11 2008 at the suggestion of Franklin T. Adams-Watters

A294201 Irregular triangle read by rows: T(n,k) is the number of k-partitions of {1..3n} that are invariant under a permutation consisting of n 3-cycles (1 <= k <= 3n).

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 2, 0, 1, 1, 3, 10, 12, 3, 9, 3, 0, 1, 1, 7, 33, 59, 30, 67, 42, 6, 18, 4, 0, 1, 1, 15, 106, 270, 216, 465, 420, 120, 235, 100, 10, 30, 5, 0, 1, 1, 31, 333, 1187, 1365, 3112, 3675, 1596, 2700, 1655, 330, 605, 195, 15, 45, 6, 0, 1

Offset: 1

Andrew Howroyd, Oct 24 2017

T(n,k) = coefficient of x^k for A(3,n)(x) in Gilbert and Riordan's article. - Robert A. Russell, Jun 13 2018

			Triangle begins:
  1,  0,   1;
  1,  1,   3,   2,   0,   1;
  1,  3,  10,  12,   3,   9,   3,   0,   1;
  1,  7,  33,  59,  30,  67,  42,   6,  18,   4,  0,  1;
  1, 15, 106, 270, 216, 465, 420, 120, 235, 100, 10, 30, 5, 0, 1;
  ...
Case n=2: Without loss of generality the permutation of two 3-cycles can be taken as (123)(456). The second row is [1, 1, 3, 2, 0, 1] because the set partitions that are invariant under this permutation in increasing order of number of parts are {{1, 2, 3, 4, 5, 6}}; {{1, 2, 3}, {4, 5, 6}}; {{1, 4}, {2, 5}, {3, 6}}, {{1, 5}, {2, 6}, {3, 4}}, {{1, 6}, {2, 4}, {3, 5}}; {{1, 2, 3}, {4}, {5}, {6}}, {{1}, {2}, {3}, {4, 5, 6}}, {{1}, {2}, {3}, {4}, {5}, {6}}.

Andrew Howroyd, Table of n, a(n) for n = 1..1395
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Row sums are A002874.
Column k=3 gives A053156.
Maximum row values are A294202.
Unrelated to A002875.
Cf. A002872, A002873, A293181.

T:= proc(n, k) option remember; `if`([n, k]=[0, 0], 1, 0)+
     `if`(n>0 and k>0, k*T(n-1, k)+T(n-1, k-1)+T(n-1, k-3), 0)
    end:
seq(seq(T(n, k), k=1..3*n), n=1..8);  # Alois P. Heinz, Sep 20 2019

T[n_, k_] := T[n,k] = If[n>0 && k>0, k T[n-1,k] + T[n-1,k-1] + T[n-1,k-3], Boole[n==0 && k==0]] (* modification of Gilbert & Riordan recursion *)
Table[T[n, k], {n,1,10}, {k,1,3n}] // Flatten (* Robert A. Russell, Jun 13 2018 *)

\\ see A056391 for Polya enumeration functions
T(n,k)={my(ci=PermCycleIndex(CylinderPerms(3,n)[2])); StructsByCycleIndex(ci,k) - if(k>1,StructsByCycleIndex(ci,k-1))}
for (n=1, 6, for(k=1, 3*n, print1(T(n,k), ", ")); print);

G(n)={Vec(-1+serlaplace(exp(sumdiv(3, d, y^d*(exp(d*x + O(x*x^n))-1)/d))))}
{ my(A=G(6)); for(n=1, #A, print(Vecrev(A[n]/y))) } \\ Andrew Howroyd, Sep 20 2019

T(n,k) = [n==0 & k==0] + [n>0 & k>0] * (k*T(n-1,k) + T(n-1,k-1) + T(n-1,k-3)). - Robert A. Russell, Jun 13 2018

T(n,k) = n!*[x^n*y^k] exp(Sum_{d|3} y^d*(exp(d*x) - 1)/d). - Andrew Howroyd, Sep 20 2019

A294202 The maximal number of partitions of {1..3n} that are invariant under a permutation consisting of n 3-cycles, and which have the same number of nonempty parts.

Original entry on oeis.org

1, 1, 3, 12, 67, 465, 3675, 30024, 299250, 3417690, 38983966, 446295630, 6494597538, 95113861987, 1365645758568, 20909896016688, 373941213111567, 6583031224561656, 114432377809889706, 2158725804226303597, 45003872172663258463, 928103099363098553160

Offset: 0

Andrew Howroyd, Oct 24 2017

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200

Maximum row values of A294201.

Cf. A002873.

G(n)={Vec(serlaplace(exp(sumdiv(3, d, y^d*(exp(d*x + O(x*x^n))-1)/d))))}
seq(n)={my(A=G(n)); vector(#A, n, vecmax(Vec(A[n])))} \\ Andrew Howroyd, Sep 20 2019

a(0)=1 prepended by Andrew Howroyd, Sep 20 2019

A036076 Expansion of e.g.f. exp((exp(p*x)-p-1)/p+exp(x)) for p=6.

Original entry on oeis.org

1, 2, 11, 87, 844, 9599, 125545, 1854234, 30407763, 546409567, 10654642428, 223763443039, 5030118977041, 120393730088818, 3054106291046267, 81792080931311015, 2304639285452820684, 68117438479292896255

Offset: 0

N. J. A. Sloane

nonn

T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
T. S. Motzkin, Sorting numbers ...: for a link to an annotated scanned version of this paper see A000262.

Robert Israel, Table of n, a(n) for n = 0..437
Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.
Index entries for sequences related to sorting

Cf. A001861, A002872, A002873, A002874, A002875, A036074, ...

egf:=  exp((exp(6*x)-6-1)/6+exp(x)):
S:= series(egf,x,501):
seq(coeff(S,x,i)*i!, i=0..20); # Robert Israel, Nov 27 2022

mx = 16; p = 6; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
Table[Sum[Binomial[n,k] * 6^k * BellB[k, 1/6] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 29 2022 *)

a(n) ~ exp(exp(p*r)/p + exp(r) - 1 - 1/p - n) * (n/r)^(n + 1/2) / sqrt((1 + p*r)*exp(p*r) + (1 + r)*exp(r)), where r = LambertW(p*n)/p - 1/(1 + p/LambertW(p*n) + n^(1 - 1/p) * (1 + LambertW(p*n)) * (p/LambertW(p*n))^(2 - 1/p)) for p=6. - Vaclav Kotesovec, Jul 03 2022

a(n) ~ (6*n/LambertW(6*n))^n * exp(n/LambertW(6*n) + (6*n/LambertW(6*n))^(1/6) - n - 7/6) / sqrt(1 + LambertW(6*n)). - Vaclav Kotesovec, Jul 10 2022

Edited by N. J. A. Sloane, Jul 11 2008 at the suggestion of Franklin T. Adams-Watters.

A036081 The number of partitions of {1..(11n)} that are invariant under a permutation consisting of n 11-cycles.

Original entry on oeis.org

1, 2, 16, 202, 3044, 52794, 1055260, 24081754, 615896308, 17347970202, 531721375308, 17595339114554, 624882463734756, 23691503493287738, 954301756159098172, 40665568780962213530, 1826521141853468785364

Offset: 0

N. J. A. Sloane

nonn

Original name: Sorting numbers.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers
Index entries for sequences related to sorting

Cf. A001861, A002872, A002873, A002874, A002875, A036074.

Column 11 of A162663.

u[0, j_] := 1; u[k_, j_] := u[k, j] = Sum[Binomial[k-1, i-1]Plus@@(u[k-i, j]#^(i-1)&/@Divisors[j]), {i, k}]; Table[u[n, 11], {n, 0, 30}] (* Vincenzo Librandi, Dec 12 2012 - after Wouter Meeussen in similar sequences *)
mx = 16; p = 11; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
Table[Sum[Binomial[n,k] * 11^k * BellB[k, 1/11] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 29 2022 *)

E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=11.
a(n) ~ exp(exp(p*r)/p + exp(r) - 1 - 1/p - n) * (n/r)^(n + 1/2) / sqrt((1 + p*r)*exp(p*r) + (1 + r)*exp(r)), where r = LambertW(p*n)/p - 1/(1 + p/LambertW(p*n) + n^(1 - 1/p) * (1 + LambertW(p*n)) * (p/LambertW(p*n))^(2 - 1/p)) for p=11. - Vaclav Kotesovec, Jul 03 2022
a(n) ~ (11*n/LambertW(11*n))^n * exp(n/LambertW(11*n) + (11*n/LambertW(11*n))^(1/11) - n - 12/11) / sqrt(1 + LambertW(11*n)). - Vaclav Kotesovec, Jul 10 2022

New name from Danny Rorabaugh, Oct 24 2015

A036079 E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=9.

Original entry on oeis.org

1, 2, 14, 150, 1942, 29174, 505318, 9957798, 219177942, 5303780758, 139554619206, 3962202725254, 120644298135478, 3918518255860342, 135117086088186662, 4925731652244913766, 189170325211554345366, 7629758975467859662678, 322296334808561664346886

Offset: 0

N. J. A. Sloane

nonn

T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
T. S. Motzkin, Sorting numbers ...: for a link to an annotated scanned version of this paper see A000262.

Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.
Index entries for sequences related to sorting

Cf. A001861, A002872, A002873, A002874, A002875, A036074...

mx = 16; p = 9; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
Table[Sum[Binomial[n,k] * 9^k * BellB[k, 1/9] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 29 2022 *)

a(n) ~ exp(exp(p*r)/p + exp(r) - 1 - 1/p - n) * (n/r)^(n + 1/2) / sqrt((1 + p*r)*exp(p*r) + (1 + r)*exp(r)), where r = LambertW(p*n)/p - 1/(1 + p/LambertW(p*n) + n^(1 - 1/p) * (1 + LambertW(p*n)) * (p/LambertW(p*n))^(2 - 1/p)) for p=9. - Vaclav Kotesovec, Jul 03 2022

a(n) ~ (9*n/LambertW(9*n))^n * exp(n/LambertW(9*n) + (9*n/LambertW(9*n))^(1/9) - n - 10/9) / sqrt(1 + LambertW(9*n)). - Vaclav Kotesovec, Jul 10 2022

Edited by N. J. A. Sloane, Jul 11 2008 at the suggestion of Franklin T. Adams-Watters.

A036080 E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=10.

Original entry on oeis.org

1, 2, 15, 175, 2452, 39703, 741177, 15771270, 375485507, 9837064575, 280338965720, 8623355105347, 284589703065137, 10022926411599482, 374900187362983015, 14830483377507515247, 618219446355189917804, 27071966121397255354079, 1241912851303663452150377

Offset: 0

N. J. A. Sloane

nonn

T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
T. S. Motzkin, Sorting numbers ...: for a link to an annotated scanned version of this paper see A000262.

Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.
Index entries for sequences related to sorting

Cf. A001861, A002872, A002873, A002874, A002875, A036074, ...

mx = 16; p = 10; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
Table[Sum[Binomial[n,k] * 10^k * BellB[k, 1/10] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 29 2022 *)

a(n) ~ exp(exp(p*r)/p + exp(r) - 1 - 1/p - n) * (n/r)^(n + 1/2) / sqrt((1 + p*r)*exp(p*r) + (1 + r)*exp(r)), where r = LambertW(p*n)/p - 1/(1 + p/LambertW(p*n) + n^(1 - 1/p) * (1 + LambertW(p*n)) * (p/LambertW(p*n))^(2 - 1/p)) for p=10. - Vaclav Kotesovec, Jul 03 2022

a(n) ~ (10*n/LambertW(10*n))^n * exp(n/LambertW(10*n) + (10*n/LambertW(10*n))^(1/10) - n - 11/10) / sqrt(1 + LambertW(10*n)). - Vaclav Kotesovec, Jul 10 2022

Edited by N. J. A. Sloane, Jul 11 2008 at the suggestion of Franklin T. Adams-Watters.

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A002872 Number of partitions of {1..2n} that are invariant under a permutation consisting of n 2-cycles.

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A293181 Irregular triangle read by rows: T(n,k) is the number of k-partitions of {1..2n} that are invariant under a permutation consisting of n 2-cycles (1 <= k <= 2n).

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A002875 Sorting numbers (see Motzkin article for details).

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A036074 Expansion of e.g.f. exp((exp(p*x) - p - 1)/p + exp(x)) for p=4.

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A294201 Irregular triangle read by rows: T(n,k) is the number of k-partitions of {1..3n} that are invariant under a permutation consisting of n 3-cycles (1 <= k <= 3n).

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A294202 The maximal number of partitions of {1..3n} that are invariant under a permutation consisting of n 3-cycles, and which have the same number of nonempty parts.

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A036076 Expansion of e.g.f. exp((exp(p*x)-p-1)/p+exp(x)) for p=6.

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