A152666
Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k runs of odd entries (1<=k<=ceiling(n/2)). For example, the permutation 321756498 has 3 runs of odd entries: 3, 175 and 9.
Original entry on oeis.org
1, 2, 4, 2, 12, 12, 36, 72, 12, 144, 432, 144, 576, 2592, 1728, 144, 2880, 17280, 17280, 2880, 14400, 115200, 172800, 57600, 2880, 86400, 864000, 1728000, 864000, 86400, 518400, 6480000, 17280000, 12960000, 2592000, 86400, 3628800, 54432000
Offset: 1
T(3,2)=2 because we have 123 and 321.
T(4,2)=12 because we have 1234, 1432, 3214, 3412, 1243, 3241 and their reverses.
Triangle starts:
1;
2;
4,2;
12,12;
36,72,12;
144,432,144;
576,2592,1728,144.
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ae := proc (n, k) options operator, arrow: factorial(n)^2*binomial(n+1, k)*binomial(n-1, k-1) end proc: ao := proc (n, k) options operator, arrow: factorial(n)*factorial(n+1)*binomial(n, k-1)*binomial(n+1, k) end proc: T := proc (n, k) if `mod`(n, 2) = 0 then ae((1/2)*n, k) else ao((1/2)*n-1/2, k) end if end proc: for n to 12 do seq(T(n, k), k = 1 .. ceil((1/2)*n)) end do; # yields sequence in triangular form
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T[n_?EvenQ, k_] := (n/2)!^2*Binomial[n/2 - 1, k - 1]*Binomial[n/2 + 1, k]; T[n_?OddQ, k_] := ((n - 1)/2 + 1)!*((n - 1)/2)!*Binomial[(n - 1)/2 + 1, k]*Binomial[(n - 1)/2, k - 1]; Table[T[n, k], {n, 1, 12}, {k, 1, Floor[(n + 1)/2]}] // Flatten (* Jean-François Alcover, Nov 13 2016 *)
A152668
Number of runs of even entries in all permutations of {1,2,...,n} (the permutation 274831659 has 3 runs of even entries: 2, 48 and 6).
Original entry on oeis.org
2, 6, 36, 192, 1440, 10800, 100800, 967680, 10886400, 127008000, 1676505600, 22992076800, 348713164800, 5492232345600, 94152554496000, 1673823191040000, 32011868528640000, 633834996867072000, 13380961044971520000
Offset: 2
a(3) = 6 because each of the permutations 123, 132, 213, 231, 312, 321 has exactly 1 run of even entries.
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ae := proc (n) options operator, arrow: (1/2)*factorial(2*n)*(n+1) end proc: ao := proc (n) options operator, arrow: n*(n+2)*factorial(2*n) end proc: a := proc (n) if `mod`(n, 2) = 0 then ae((1/2)*n) else ao((1/2)*n-1/2) end if end proc; seq(a(n), n = 2 .. 20);
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a[n_] := If[EvenQ[n], (n/2+1)n!/2, ((n-1)/2)((n-1)/2+2)(n-1)!];
Table[a[n], {n, 2, 20}] (* Jean-François Alcover, Apr 09 2024 *)
A152873
Number of permutations of {1,2,...,n} (n>=2) having a single run of even entries. For example, the permutation 513284679 has a single run of even entries: 2846.
Original entry on oeis.org
2, 6, 12, 48, 144, 720, 2880, 17280, 86400, 604800, 3628800, 29030400, 203212800, 1828915200, 14631321600, 146313216000, 1316818944000, 14485008384000, 144850083840000, 1738201006080000, 19120211066880000, 248562743869440000, 2982752926433280000, 41758540970065920000
Offset: 2
a(4) = 12 because we have 2413, 2431, 4213, 4231, 1243, 1423 and their reverses.
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ae := proc (n) options operator, arrow: factorial(n)^2*(n+1) end proc: ao := proc (n) options operator, arrow: factorial(n)*factorial(n+2) end proc: a := proc (n) if `mod`(n, 2) = 0 then ae((1/2)*n) else ao((1/2)*n-1/2) end if end proc; seq(a(n), n = 2 .. 23);
# second Maple program:
a:= n-> (h-> h!*(h+1+(n mod 2))!)(iquo(n, 2)):
seq(a(n), n=2..25); # Alois P. Heinz, Sep 24 2024
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a[n_] := If[OddQ[n], ((n - 1)/2)!*((n + 3)/2)!, (n/2 + 1) ((n/2)!)^2]; Array[a, 25, 2] (* Amiram Eldar, Jan 22 2023 *)
Showing 1-3 of 3 results.
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