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.

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

Offset: 0

Author

Wouter Meeussen, Mar 15 2003

Keywords

• nonn

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

Number of achiral color patterns in a row or loop of length n. Two color patterns are equivalent if the colors are permuted. - Robert A. Russell, Apr 23 2018

Also the number of self-complementary set partitions of {1, ..., n}. The complement of a set partition pi of {1, ..., n} is defined as n + 1 - pi (elementwise) on page 3 of Callan. For example, the complement of {{1,5},{2},{3,6},{4}} is {{1,4},{2,6},{3},{5}}. - Gus Wiseman, Feb 13 2019

Examples

			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.
For a(4)=7, the row patterns are AAAA, AABB, ABAB, ABBA, ABBC, ABCA, and ABCD (same as previous example).  The loop patterns are AAAA, AAAB, AABB, AABC, ABAB, ABAC, and ABCD. - _Robert A. Russell_, Apr 23 2018
From _Gus Wiseman_, Feb 13 2019: (Start)
The a(1) = 1 through a(5) = 12 self-complementary set partitions:
  {{1}}  {{12}}    {{123}}      {{1234}}        {{12345}}
         {{1}{2}}  {{13}{2}}    {{12}{34}}      {{1245}{3}}
                   {{1}{2}{3}}  {{13}{24}}      {{135}{24}}
                                {{14}{23}}      {{15}{234}}
                                {{1}{23}{4}}    {{1}{234}{5}}
                                {{14}{2}{3}}    {{12}{3}{45}}
                                {{1}{2}{3}{4}}  {{135}{2}{4}}
                                                {{14}{25}{3}}
                                                {{15}{24}{3}}
                                                {{1}{24}{3}{5}}
                                                {{15}{2}{3}{4}}
                                                {{1}{2}{3}{4}{5}}
(End)
		

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
• Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See p. 18.
• David Callan, On conjugates for set partitions and integer compositions, arXiv:math/0508052 [math.CO], 2005.
• Juan B. Gil and Luiz E. Lopez, Enumeration of symmetric arc diagrams, arXiv:2203.10589 [math.CO], 2022.
• S. V. Pemmaraju and S. S. Skiena, The New Combinatorica, 2001.
• Frank Ruskey, Set Partitions

Crossrefs

Cf. A002872, A080337.

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

Cf. A000126, A000296, A124323, A169985, A324012, A324013, A324014.

Programs

Mathematica

< 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-François Alcover, Feb 29 2016, after Paul D. Hanna *)
(* Ach[n, k] is the number of achiral color patterns for a row or loop of n
  colors containing exactly k different colors *)
Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0],
  k Ach[n-2, k] + Ach[n-2, k-1] + Ach[n-2, k-2]]
Table[Sum[Ach[n, k], {k, 0, n}], {n, 0, 30}] (* Robert A. Russell, Apr 23 2018 *)
x[n_] := x[n] = If[n < 2, n+1, 2x[n-1] + (n-1)x[n-2]]; (* A005425 *)
Table[Sum[StirlingS2[Ceiling[n/2], k] x[k-Mod[n, 2]], {k, 0, Ceiling[n/2]}],
  {n, 0, 30}] (* Robert A. Russell, Apr 27 2018, after Knuth reference *)

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

From Robert A. Russell, Apr 23 2018: (Start)

a(n) = Sum_{k=0..n} Ach(n,k) where

Ach(n,k) = [n>1]*(k*Ach(n-2,k)+Ach(n-2,k-1)+Ach(n-2,k-2)) + [n<2]*[n==k]*[n>=0].

a(n) = 2*A103293(n+1) - A000110(n). (End)

a(n) = [n==0 mod 2]*Sum_{k=0..n/2} Stirling2(n/2, k)*A005425(k) + [n==1 mod 2] * Sum_{k=1..(n+1)/2} Stirling2((n+1)/2, k) * A005425(k-1). (from Knuth reference)

a(n) = 2*A084708(n) - A084423(n). - Robert A. Russell, Apr 27 2018

Extensions

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