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A080107 Number of fixed points of permutation of SetPartitions under {1,2,..,n}->{n,n-1,..,1}. Number of symmetric arrangements of non-attacking rooks on upper half of n X n chessboard. 8
1, 1, 2, 3, 7, 12, 31, 59, 164, 339, 999, 2210, 6841, 16033, 51790, 127643, 428131, 1103372, 3827967, 10269643, 36738144, 102225363, 376118747, 1082190554, 4086419601, 12126858113, 46910207114, 143268057587, 566845074703, 1778283994284, 7186474088735 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Even-numbered terms a(2k) are A002872: 2,7,31,164,999 ("Sorting numbers"); odd-numbered terms are its binomial transform, A080337. The symmetrical set partitions of {-n,...,-1,0,1,...,n} can be classified by the partition containing 0. Thus we get the sum over k of {n choose k} times the number of symmetrical set partitions of 2n-2k elements. - Don Knuth, Nov 23 2003

Number of partitions of n numbers that are symmetrical and cannot be nested (i.e. include a pattern of the form abab). - Douglas Boffey, May 21 2015

REFERENCES

D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.5 (p. 765).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

S. V. Pemmaraju and S. S. Skiena, NewCombinatorica

Frank Ruskey, Set Partitions

FORMULA

Knuth gives recurrences and generating functions.

a(n) = Sum_{k=0..t(n)} (-1)^k*A125810(n,k) where A125810 is a triangle of coefficients for a q-analog of the Bell numbers and t(n)=A125811(n)-1. - Paul D. Hanna, Jan 19 2009

EXAMPLE

Of the set partitions of 4, the following 7 are invariant under 1->4, 2->3, 3->2, 4->1: {{1,2,3,4}}, {{1,2},{3,4}}, {{1,4},{2,3}}, {{1,3},{2,4}}, {{1},{2,3},{4}}, {{1,4},{2},{3}}, {{1},{2},{3},4}}, so a[4]=7.

MATHEMATICA

<<DiscreteMath`NewCombinatorica`; Table[t = SetPartitions[n]; t= t /. Thread[Range[n] -> Range[n, 1, -1]]; t= 1 + RankSetPartition /@ t; t= ToCycles[t]; t= Cases[t, {_Integer}]; Length[t], {n, 7}]

(* second program: *)

QB[n_, q_] := QB[n, q] = Sum[QB[j, q] QBinomial[n-1, j, q], {j, 0, n-1}] // FunctionExpand // Simplify; QB[0, q_]=1; QB[1, q_]=1; Table[cc = CoefficientList[QB[n, q], q]; cc.Table[(-1)^(k+1), {k, 1, Length[cc]}], {n, 0, 30}] (* Jean-Fran├žois Alcover, Feb 29 2016, after Paul D. Hanna *)

CROSSREFS

Cf. A002872, A080337.

Cf. A125810. - Paul D. Hanna, Jan 19 2009

Sequence in context: A100982 A186009 A034786 * A056156 A112837 A056355

Adjacent sequences:  A080104 A080105 A080106 * A080108 A080109 A080110

KEYWORD

nonn

AUTHOR

Wouter Meeussen, Mar 15 2003

EXTENSIONS

Offset set to 0 by Alois P. Heinz, May 23 2015

STATUS

approved

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Last modified July 22 00:13 EDT 2017. Contains 289648 sequences.