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A002872 Number of partitions of {1..2n} that are invariant under a permutation consisting of n 2-cycles.
(Formerly M1786 N0705)
26
1, 2, 7, 31, 164, 999, 6841, 51790, 428131, 3827967, 36738144, 376118747, 4086419601, 46910207114, 566845074703, 7186474088735, 95318816501420, 1319330556537631, 19013488408858761, 284724852032757686, 4422344774431494155, 71125541977466879231 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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
REFERENCES
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).
LINKS
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.
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=4SYM, arXiv:1010.1683 [hep-th], 2010, (E.2).
J. Pasukonis and S. Ramgoolam, From counting to construction for BPS states in N=4SYM, 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.
FORMULA
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
EXAMPLE
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
MAPLE
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
MATHEMATICA
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 *)
CROSSREFS
u[n,j] is A162663.
Row sums of A293181.
Column k=2 of A306024.
Cf. A005425.
Sequence in context: A007446 A277396 A227119 * A105216 A260532 A193657
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
Edited by Franklin T. Adams-Watters, Jul 09 2009
STATUS
approved

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Last modified April 21 17:27 EDT 2024. Contains 371874 sequences. (Running on oeis4.)