

A114938


Number of permutations of the multiset {1,1,2,2,...,n,n} with no two consecutive terms equal.


24



1, 0, 2, 30, 864, 39480, 2631600, 241133760, 29083420800, 4467125013120, 851371260364800, 197158144895712000, 54528028997584665600, 17752366094818747392000, 6720318485119046923315200, 2927066537906697348594432000, 1453437879238150456164433920000
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OFFSET

0,3


COMMENTS

a(n) is also the number of (0,1)matrices A=(a_ij) of size n X 2n such that each row has exactly two 1's and each column has exactly one 1 and with the restriction that no 1 stands on the line from a_11 to a_22.  Shanzhen Gao, Feb 24 2010
a(n) is the number of permutations of the multiset {1,1,2,2,...,n,n} with no fixed points.  Alexander Burstein, May 16 2020
Also the number of 2uniform ordered set partitions of {1...2n} containing no two successive vertices in the same block.  Gus Wiseman, Jul 04 2020


REFERENCES

R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997. Chapter 2, Sieve Methods, Example 2.2.3, page 68.


LINKS



FORMULA

a(n) = Sum_{k=0..n} ((binomial(n, k)*(1)^(nk)*(n+k)!)/2^k).


EXAMPLE

a(2) = 2 because there are two permutations of {1,1,2,2} avoiding equal consecutive terms: 1212 and 2121.


MATHEMATICA

Table[Sum[Binomial[n, i](2ni)!/2^(ni) (1)^i, {i, 0, n}], {n, 0, 20}] (* Geoffrey Critzer, Jan 02 2013, and adapted to the extension by Stefano Spezia, Nov 15 2018 *)
Table[Length[Select[Permutations[Join[Range[n], Range[n]]], !MatchQ[#, {___, x_, x_, ___}]&]], {n, 0, 5}] (* Gus Wiseman, Jul 04 2020 *)


PROG

(PARI) vector(20, n, sum(k=0, n, binomial(n, k)*(1)^(nk)*(n+k)!/2^k)) \\ Michel Marcus, Aug 10 2015
(Magma) I:=[0, 2]; [n le 2 select I[n] else n*(2*n1)*Self(n1) + (n1)*n*Self(n2): n in [1..20]]; // Vincenzo Librandi, Aug 10 2015


CROSSREFS

Cf. A114939 = preferred seating arrangements of n couples.
Cf. A007060 = arrangements of n couples with no adjacent spouses; A007060(n) = 2^n * A114938(n) (this sequence).
Cf. A278990 = number of loopless linear chord diagrams with n chords.
Cf. A000806 = Bessel polynomial y_n(1).
The version for multisets with prescribed multiplicities is A335125.
The version for prime indices is A335452.
Antirun compositions are counted by A003242.
Antirun compositions are ranked by A333489.
Inseparable partitions are counted by A325535.
Inseparable partitions are ranked by A335448.
Separable partitions are counted by A325534.
Separable partitions are ranked by A335433.
Other sequences involving the multiset {1,1,2,2,...,n,n}: A001147, A007717, A020555, A094574, A316972.


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



