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 A177248 Triangle read by rows: T(n,k) is the number of permutations of [n] having k adjacent transpositions (0 <= k <= floor(n/2)). An adjacent transposition is a cycle of the form (i, i+1). 5
 1, 1, 1, 1, 4, 2, 19, 4, 1, 99, 18, 3, 611, 99, 9, 1, 4376, 612, 48, 4, 35621, 4376, 306, 16, 1, 324965, 35620, 2190, 100, 5, 3285269, 324965, 17810, 730, 25, 1, 36462924, 3285270, 162480, 5940, 180, 6, 440840359, 36462924, 1642635, 54160, 1485, 36, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Row n contains 1 + floor(n/2) entries. Sum of entries in row n = n! (A000142). T(n,0) = A177249(n). Sum_{k>=0} k*a(n,k) = (n-1)! (n >= 2). LINKS Table of n, a(n) for n=0..48. R. A. Brualdi and E. Deutsch, Adjacent q-cycles in permutations, arXiv:1005.0781 [math.CO], 2010. FORMULA T(n,k) = Sum_{j=0..floor(n/2)} (-1)^(k+j)*binomial(j,k)*(n-j)!/j!. EXAMPLE T(5,2)=3 because we have (12)(34)(5), (12)(3)(45), and (1)(23)(45). Triangle starts: 1; 1; 1, 1; 4, 2; 19, 4, 1; 99, 18, 3; 611, 99, 9, 1; MAPLE T := proc (n, k) options operator, arrow: sum((-1)^(k+j)*binomial(j, k)*factorial(n-j)/factorial(j), j = 0 .. floor((1/2)*n)) end proc: for n from 0 to 12 do seq(T(n, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form MATHEMATICA T[n_, k_] := Sum[(-1)^(k + j)*Binomial[j, k]*(n - j)!/j!, {j, 0, n/2}]; Table[T[n, k], {n, 0, 12}, {k, 0, n/2}] // Flatten (* Jean-François Alcover, Nov 20 2017 *) PROG (PARI) T(n, k) = sum(j=0, n\2, (-1)^(k+j)*binomial(j, k)*(n-j)!/j!); tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n, k), ", ")); print); \\ Michel Marcus, Nov 21 2017 CROSSREFS Cf. A000166, A008290, A177249, A177250, A177251, A177252, A177253. Sequence in context: A117692 A052966 A305135 * A191706 A081797 A354094 Adjacent sequences: A177245 A177246 A177247 * A177249 A177250 A177251 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, May 07 2010 STATUS approved

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Last modified June 8 15:33 EDT 2023. Contains 363165 sequences. (Running on oeis4.)