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).

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

Offset: 0

Author

Emeric Deutsch, May 07 2010

Keywords

• nonn
• tabf

Comments

Row n contains 1 + floor(n/2) entries.

Sum of entries in row n = n! (A000142).

Examples

			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;
		

Links

• Seiichi Manyama, Rows n = 0..200, flattened
• R. A. Brualdi and E. Deutsch, Adjacent q-cycles in permutations, arXiv:1005.0781 [math.CO], 2010.

Crossrefs

Columns k=0..3 give A177249, A370524, A370426, A370529.

Cf. A000166, A008290, A177249, A177250, A177251, A177252, A177253.

Cf. A000142 (row sums).

Programs

Magma

F:=Factorial;
A177248:= func< n,k | (&+[(-1)^j*F(n-k-j)/(F(k)*F(j)): j in [0..Floor((n-2*k)/2)]]) >;
[A177248(n,k): k in [0..Floor(n/2)], n in [0..16]]; // G. C. Greubel, Apr 28 2024

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 *)

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

SageMath

f=factorial;
def A177248(n,k): return sum((-1)^j*f(n-k-j)/(f(k)*f(j)) for j in range(1+(n-2*k)//2))
flatten([[A177248(n,k) for k in range(1+n//2)] for n in range(17)]) # G. C. Greubel, Apr 28 2024

Formula

T(n, k) = Sum_{j=0..floor(n/2)} (-1)^(k+j)*binomial(j,k)*(n-j)!/j!.

T(n, 0) = A177249(n).

Sum_{k=0..floor(n/2)} k*T(n,k) = (n-1)! (n >= 2).

G.f. of column k: (1/k!) * Sum_{j>=k} j! * x^(j+k) / (1+x^2)^(j+1). - Seiichi Manyama, Feb 24 2024