A306024 -id:A306024 - 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

A305962 Number A(n,k) of length-n restricted growth strings (RGS) with growth <= k and fixed first element; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 4, 12, 15, 1, 1, 1, 5, 22, 59, 52, 1, 1, 1, 6, 35, 150, 339, 203, 1, 1, 1, 7, 51, 305, 1200, 2210, 877, 1, 1, 1, 8, 70, 541, 3125, 10922, 16033, 4140, 1, 1, 1, 9, 92, 875, 6756, 36479, 110844, 127643, 21147, 1

Offset: 0

Alois P. Heinz, Jun 15 2018

A(n,k) counts strings [s_1, ..., s_n] with 1 = s_1 <= s_i <= k + max_{j

			A(0,2) = 1: the empty string.
A(1,2) = 1: 1.
A(2,2) = 3: 11, 12, 13.
A(3,2) = 12: 111, 112, 113, 121, 122, 123, 124, 131, 132, 133, 134, 135.
Square array A(n,k) begins:
  1,   1,     1,      1,      1,       1,       1,       1, ...
  1,   1,     1,      1,      1,       1,       1,       1, ...
  1,   2,     3,      4,      5,       6,       7,       8, ...
  1,   5,    12,     22,     35,      51,      70,      92, ...
  1,  15,    59,    150,    305,     541,     875,    1324, ...
  1,  52,   339,   1200,   3125,    6756,   12887,   22464, ...
  1, 203,  2210,  10922,  36479,   96205,  216552,  435044, ...
  1, 877, 16033, 110844, 475295, 1530025, 4065775, 9416240, ...

Alois P. Heinz, Antidiagonals n = 0..150, flattened

Columns k=0-10 give: A000012, A000110, A080337, A189845, A305964, A305965, A305966, A305967, A305968, A305969, A305970.
Main diagonal gives: A305963.
Antidiagonal sums give: A305971.
Cf. A306024.

b:= proc(n, k, m) option remember; `if`(n=0, 1,
      add(b(n-1, k, max(m, j)), j=1..m+k))
    end:
A:= (n, k)-> b(n, k, 1-k):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
A:= (n, k)-> `if`(n=0, 1, (n-1)!*coeff(series(exp(x+add(
              (exp(j*x)-1)/j, j=1..k)), x, n), x, n-1)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[n_, k_, m_] := b[n, k, m] = If[n==0, 1, Sum[b[n-1, k, Max[m, j]], {j, 1, m+k}]];
A[n_, k_] := b[n, k, 1-k];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 27 2019, after Alois P. Heinz *)

A(n,k) = (n-1)! * [x^(n-1)] exp(x+Sum_{j=1..k} (exp(j*x)-1)/j) for n>0, A(0,k) = 1.

A306027 Number of length-n restricted growth strings (RGS) with growth <= three and first element in [3].

Original entry on oeis.org

1, 3, 15, 95, 717, 6221, 60619, 652595, 7667957, 97415437, 1328078067, 19310628827, 297932883437, 4856255827013, 83315165338923, 1499606140412403, 28237439054261893, 554849548137840189, 11351488777441797187, 241314920928367232747, 5320846918247724517117

Offset: 0

Alois P. Heinz, Jun 17 2018

nonn

			a(2) = 15: 11, 12, 13, 14, 21, 22, 23, 24, 25, 31, 32, 33, 34, 35, 36.

Alois P. Heinz, Table of n, a(n) for n = 0..481

Column k=3 of A306024.

Cf. A189845.

b:= proc(n, m) option remember; `if`(n=0, 1,
      add(b(n-1, max(m, j)), j=1..m+3))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);
# second Maple program:
a:= n-> n!*coeff(series(exp(add((exp(j*x)-1)/j, j=1..3)), x, n+1), x, n):
seq(a(n), n=0..25);

E.g.f.: exp(Sum_{j=1..3} (exp(j*x)-1)/j).

A306028 Number of length-n restricted growth strings (RGS) with growth <= four and first element in [4].

Original entry on oeis.org

1, 4, 26, 214, 2096, 23578, 297692, 4150798, 63148652, 1038505282, 18324165284, 344791676446, 6883214479964, 145160792680210, 3221934217192772, 75022325591768782, 1827426669105650588, 46448931646662304066, 1229212096172351894852, 33800410177112965182910

Offset: 0

Alois P. Heinz, Jun 17 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..459

Column k=4 of A306024.

Cf. A305964.

b:= proc(n, m) option remember; `if`(n=0, 1,
      add(b(n-1, max(m, j)), j=1..m+4))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);
# second Maple program:
a:= n-> n!*coeff(series(exp(add((exp(j*x)-1)/j, j=1..4)), x, n+1), x, n):
seq(a(n), n=0..25);

E.g.f.: exp(Sum_{j=1..4} (exp(j*x)-1)/j).

A306025 Number of length-n restricted growth strings (RGS) with growth <= n and first element in [n].

Original entry on oeis.org

1, 1, 7, 95, 2096, 67354, 2943277, 166862583, 11858631472, 1029154793775, 106837050484924, 13046411412001307, 1848336205780389404, 300289842081446066173, 55393980428260038660617, 11503469972529028999979343, 2669299049110696359069533376

Offset: 0

Alois P. Heinz, Jun 17 2018

nonn

			a(0) = 1: the empty string.
a(1) = 1: 1.
a(2) = 7: 11, 12, 13, 21, 22, 23, 24.

Alois P. Heinz, Table of n, a(n) for n = 0..265

Main diagonal of A306024.

Cf. A305963.

b:= proc(n, k, m) option remember; `if`(n=0, 1,
      add(b(n-1, k, max(m, j)), j=1..m+k))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..20);
# second Maple program:
a:= n-> n!*coeff(series(exp(add((exp(j*x)-1)/j, j=1..n)), x, n+1), x, n):
seq(a(n), n=0..20);

b[n_, k_, m_] := b[n, k, m] = If[n == 0, 1,
   Sum[b[n - 1, k, Max[m, j]], {j, 1, m + k}]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Sep 07 2022, after Alois P. Heinz *)

a(n) = n! * [x^n] exp(Sum_{j=1..n} (exp(j*x)-1)/j).

a(n) = A306024(n,n).

A306029 Number of length-n restricted growth strings (RGS) with growth <= five and first element in [5].

Original entry on oeis.org

1, 5, 40, 405, 4875, 67354, 1044045, 17867125, 333554020, 6730070329, 145676361731, 3362266525430, 82326965117385, 2129349953723509, 57961263778376192, 1655067729384150829, 49437118345913831595, 1540860755766376984434, 50000885646431513577973

Offset: 0

Alois P. Heinz, Jun 17 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..444

Column k=5 of A306024.

Cf. A305965.

b:= proc(n, m) option remember; `if`(n=0, 1,
      add(b(n-1, max(m, j)), j=1..m+5))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);
# second Maple program:
a:= n-> n!*coeff(series(exp(add((exp(j*x)-1)/j, j=1..5)), x, n+1), x, n):
seq(a(n), n=0..25);

E.g.f.: exp(Sum_{j=1..5} (exp(j*x)-1)/j).

A306030 Number of length-n restricted growth strings (RGS) with growth <= six and first element in [6].

Original entry on oeis.org

1, 6, 57, 685, 9780, 160201, 2943277, 59687920, 1320233315, 31557691541, 809161436022, 22121068343155, 641530646758325, 19651776950222806, 633510644286624717, 21422880077590022265, 757789084383273607060, 27969244566731240796621, 1074750913823536151018737

Offset: 0

Alois P. Heinz, Jun 17 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..431

Column k=6 of A306024.

Cf. A305966.

b:= proc(n, m) option remember; `if`(n=0, 1,
      add(b(n-1, max(m, j)), j=1..m+6))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);
# second Maple program:
a:= n-> n!*coeff(series(exp(add((exp(j*x)-1)/j, j=1..6)), x, n+1), x, n):
seq(a(n), n=0..25);

E.g.f.: exp(Sum_{j=1..6} (exp(j*x)-1)/j).

A306031 Number of length-n restricted growth strings (RGS) with growth <= seven and first element in [7].

Original entry on oeis.org

1, 7, 77, 1071, 17689, 335083, 7117789, 166862583, 4266097185, 117856459427, 3492369757317, 110332338885167, 3697474419284233, 130877053323860507, 4875005854140483629, 190478720135836588519, 7784916624799021983345, 331981949761308979115795

Offset: 0

Alois P. Heinz, Jun 17 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..421

Column k=7 of A306024.

Cf. A305967.

b:= proc(n, m) option remember; `if`(n=0, 1,
      add(b(n-1, max(m, j)), j=1..m+7))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);
# second Maple program:
a:= n-> n!*coeff(series(exp(add((exp(j*x)-1)/j, j=1..7)), x, n+1), x, n):
seq(a(n), n=0..25);

E.g.f.: exp(Sum_{j=1..7} (exp(j*x)-1)/j).

A306032 Number of length-n restricted growth strings (RGS) with growth <= eight and first element in [8].

Original entry on oeis.org

1, 8, 100, 1580, 29632, 637220, 15363376, 408744260, 11858631472, 371738275844, 12498545568496, 447999414313412, 17033296095318832, 684009592157461508, 28904630032314771184, 1281216363971401581764, 59402574686629293902896, 2873649326399265142739972

Offset: 0

Alois P. Heinz, Jun 17 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..413

Column k=8 of A306024.

Cf. A305968.

b:= proc(n, m) option remember; `if`(n=0, 1,
      add(b(n-1, max(m, j)), j=1..m+8))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);
# second Maple program:
a:= n-> n!*coeff(series(exp(add((exp(j*x)-1)/j, j=1..8)), x, n+1), x, n):
seq(a(n), n=0..25);

E.g.f.: exp(Sum_{j=1..8} (exp(j*x)-1)/j).

Showing 1-10 of 14 results. Next

cp's OEIS Frontend

A306026 Antidiagonal sums of A306024.

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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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A305962 Number A(n,k) of length-n restricted growth strings (RGS) with growth <= k and fixed first element; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A306027 Number of length-n restricted growth strings (RGS) with growth <= three and first element in [3].

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A306028 Number of length-n restricted growth strings (RGS) with growth <= four and first element in [4].

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A306025 Number of length-n restricted growth strings (RGS) with growth <= n and first element in [n].

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A306029 Number of length-n restricted growth strings (RGS) with growth <= five and first element in [5].

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A306030 Number of length-n restricted growth strings (RGS) with growth <= six and first element in [6].

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