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Triangle read by rows: T(n,k) = one-half number of permutations of length n with exactly k rising or falling successions, for n >= 1, 0 <= k <= n-1. T(1,0) = 1 by convention.
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%I #19 Jul 26 2022 03:04:56

%S 1,0,1,0,2,1,1,5,5,1,7,20,24,8,1,45,115,128,60,11,1,323,790,835,444,

%T 113,14,1,2621,6217,6423,3599,1099,183,17,1,23811,55160,56410,32484,

%U 11060,2224,270,20,1,239653,545135,554306,325322,118484,27152,3950,374,23,1,2648395

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

%C (1/2) times number of permutations of 12...n such that exactly k of the following occur: 12, 23, ..., (n-1)n, 21, 32, ..., n(n-1).

%D F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.

%H Alois P. Heinz, <a href="/A086856/b086856.txt">Rows n = 1..141, flattened</a>

%H J. Riordan, <a href="http://dx.doi.org/10.1214/aoms/1177700181">A recurrence for permutations without rising or falling successions</a>, Ann. Math. Statist. 36 (1965), 708-710.

%e Triangle T(n,k) begins:

%e 1;

%e 0, 1;

%e 0, 2, 1;

%e 1, 5, 5, 1;

%e 7, 20, 24, 8, 1;

%e 45, 115, 128, 60, 11, 1;

%e 323, 790, 835, 444, 113, 14, 1;

%e ...

%p S:= proc(n) option remember; `if`(n<4, [1, 1, 2*t, 4*t+2*t^2]

%p [n+1], expand((n+1-t)*S(n-1) -(1-t)*(n-2+3*t)*S(n-2)

%p -(1-t)^2*(n-5+t)*S(n-3) +(1-t)^3*(n-3)*S(n-4)))

%p end:

%p T:= (n, k)-> ceil(coeff(S(n), t, k)/2):

%p seq(seq(T(n, k), k=0..n-1), n=1..10); # _Alois P. Heinz_, Jan 11 2013

%t 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]/2]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 11}] // Flatten (* _Jean-François Alcover_, Jan 14 2014, translated from _Alois P. Heinz_'s Maple code *)

%Y Diagonals give A001266 (and A002464), A000130, A000349, A001267, A001268.

%Y Triangle in A001100 divided by 2. A010028 transposed.

%Y Row sums give A001710.

%K tabl,nonn

%O 1,5

%A _N. J. A. Sloane_, Aug 19 2003