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A098813 For a string of letters of length k, say abc...def, let f(k) be the string of length k-1 consisting of the adjacent pairs ab, bc, cd, ..., de, ef. Given n, let U be the string of length 2n consisting of n 1's followed by n 2's: 11...122...2. Then a(n) is the number of the C(2n,n) permutations V of U such that f(U) and f(V) agree in exactly one place. 2
1, 1, 4, 19, 57, 178, 543, 1591, 4598, 13117, 36999, 103514, 287653, 794847, 2186054, 5988339, 16347999, 44497490, 120804023, 327217525, 884531586, 2386747391, 6429784509, 17296261734, 46465809007, 124678595953, 334173980818, 894778164125 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The number of V's such that f(U) and f(V) agree in no positions gives the A051292(n+1) sequence (Whitney numbers): 1, 4, 9, 21, 52, 127, 313, 778, 1941, 4863, 12228, 30817, ...

LINKS

Table of n, a(n) for n=1..28.

EXAMPLE

For n=1, U = 12 and only one V, 12 is a 1-match, so a(1)=1.

For n=2, U = 1122, f(U) = 11,12,22 and only one V, 2121 is a 1-match, with f(v) = 21,12,21, so a(2)=1.

For n=3, U = 111222 and only the four V's 112212, 121122, 121221 and 221211 are 1-matches, so a(3)=4.

MAPLE

with(combinat): for n from 1 to 10 do y:=0:B:=array: M:=[seq(11, i=1..n-1), seq(12, i=n), seq(22, i=n+1..2*n-1)]: S:=[seq(i, i=1..2*n)]: L:=choose(S, n): for j from 1 to binomial(2*n, n) do for k from 1 to 2*n-1 do if member(k, L[j]) then B[k]:=10 else B[k]:=20 end if: if member(k+1, L[j]) then B[k]:=B[k]+1 else B[k]:=B[k]+2 end if end do: x:=0: for l from 1 to 2*n-1 do if B[l]=M[l] then x:=x+1 end if end do: if x=1 then y:=y+1 end if end do: print(y) end do: # Miklos Kristof, Oct 07 2004

PROG

(Python)

def find(bits_in, n0, n1, match):

....global count, U

....bitsleft = n0 + n1

....if bitsleft==0:

........if match:

............count += 1

....else:

........bitsleft -= 1

........if n0 > 0:

............bits_out = bits_in<<1

............new_match = (bits_out&3) == ((U >> bitsleft)&3)

............if not (match and new_match):

................find(bits_out, n0-1, n1, match or new_match)

........if n1 > 0:

............bits_out = (bits_in<<1)|1;

............new_match = (bits_out&3) == ((U >> bitsleft)&3)

............if not (match and new_match):

................find(bits_out, n0, n1-1, match or new_match)

def A098813(n):

....global count, U

....count = 0 ; U = (1<<n)-1

....find(0, n-1, n, False)

....find(1, n, n-1, False)

....return count

# Bert Dobbelaere, Dec 23 2018

CROSSREFS

Cf. A051292.

Sequence in context: A283333 A332697 A134507 * A212039 A055485 A000306

Adjacent sequences:  A098810 A098811 A098812 * A098814 A098815 A098816

KEYWORD

nonn,nice

AUTHOR

Zerinvary Lajos (with help from Miklos Kristof), Oct 07 2004

EXTENSIONS

a(13)-a(15) from Ray Chandler, Oct 25 2004

a(16)-a(28) from Bert Dobbelaere, Dec 24 2018

STATUS

approved

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Last modified October 24 08:15 EDT 2021. Contains 348217 sequences. (Running on oeis4.)