A001100 Triangle read by rows: T(n,k) = number of permutations of length n with exactly k rising or falling successions, for n >= 1, 0 <= k <= n-1.

1, 0, 2, 0, 4, 2, 2, 10, 10, 2, 14, 40, 48, 16, 2, 90, 230, 256, 120, 22, 2, 646, 1580, 1670, 888, 226, 28, 2, 5242, 12434, 12846, 7198, 2198, 366, 34, 2, 47622, 110320, 112820, 64968, 22120, 4448, 540, 40, 2, 479306, 1090270, 1108612, 650644, 236968, 54304, 7900, 748, 46, 2

Offset: 1

Author

N. J. A. Sloane, Aug 19 2003

Keywords

• tabl
• nonn

Comments

Number of permutations of 12...n such that exactly k of the following occur: 12, 23, ..., (n-1)n, 21, 32, ..., n(n-1).

Examples

			Triangle T(n,k) begins (n >= 1, k = 0..n-1):
    1;
    0,    2;
    0,    4,    2;
    2,   10,   10,   2;
   14,   40,   48,  16,   2;
   90,  230,  256, 120,  22,  2;
  646, 1580, 1670, 888, 226, 28, 2;
  ...
		

References

• F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.
• J. Riordan, A recurrence for permutations without rising or falling successions. Ann. Math. Statist. 36 (1965), 708-710.
• David Sankoff and Lani Haque, Power Boosts for Cluster Tests, in Comparative Genomics, Lecture Notes in Computer Science, Volume 3678/2005, Springer-Verlag.

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• D. P. Robbins, The probability that neighbors remain neighbors after random rearrangements, Amer. Math. Monthly 87 (1980), 122-124.

Crossrefs

Columns give A002464, A086852, A086853, A086854, A086955.

Triangle in A086856 multiplied by 2. Cf. A010028.

Cf. A322294, A322295, A322296.

Programs

Maple

S:= proc(n) option remember; `if`(n<4, [1, 1, 2*t, 4*t+2*t^2]
       [n+1], expand((n+1-t)*S(n-1) -(1-t)*(n-2+3*t)*S(n-2)
       -(1-t)^2*(n-5+t)*S(n-3) +(1-t)^3*(n-3)*S(n-4)))
    end:
T:= (n, k)-> coeff(S(n), t, k):
seq(seq(T(n, k), k=0..n-1), n=1..10);  # Alois P. Heinz, Jan 11 2013

Mathematica

s[n_] := s[n] = If[n < 4, {1, 1, 2*t, 4*t + 2*t^2}[[n + 1]], Expand[(n + 1 - t)*s[n - 1] - (1 - t)*(n - 2 + 3*t)*s[n - 2] - (1 - t)^2*(n - 5 + t)*s[n - 3] + (1 - t)^3*(n - 3)* s[n - 4]]]; t[n_, k_] := Ceiling[Coefficient[s[n], t, k]]; Flatten[ Table[ Table[t[n, k], {k, 0, n - 1}], {n, 1, 10}]] (* Jean-François Alcover, Jan 25 2013, translated from Alois P. Heinz's Maple program *)

Formula

Let T{n, k} = number of permutations of 12...n with exactly k rising or falling successions. Let S[n](t) = Sum_{k >= 0} T{n, k}*t^k. Then S[0] = 1; S[1] = 1; S[2] = 2*t; S[3] = 4*t+2*t^2; for n >= 4, S[n] = (n+1-t)*S[n-1] - (1-t)*(n-2+3*t)*S[n-2] - (1-t)^2*(n-5+t)*S[n-3] + (1-t)^3*(n-3)*S[n-4].

T(n, 0) = n! + Sum_{i=1..m-1} (-1)^i*(n-i)!*Sum_{j=1..i} 2^j*binomial(i-1, j-1)*binomial(n-i, j), and T(n, k) = Sum_{i=1..m-1} (-1)^(i-k)*binomial(i, k)*(n-i)!*Sum_{j=1..i} 2^j*binomial(i-1, j-1)*binomial(n-i, j), for k >= 1, and n >= 1. See the D. P.Robbins link for A(n, k) = T(n, k), and his comment concerning the case k = i = 0 . - Wolfdieter Lang, May 17 2025