A220183 -id:A220183 - cp's OEIS frontend

A242783 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive step pattern given by the binary expansion of n, where 1=up and 0=down; triangle T(n,k), n>=0, read by rows.

1, 1, 2, 5, 1, 21, 3, 70, 50, 450, 270, 4326, 602, 99, 12, 1, 34944, 5376, 209863, 139714, 13303, 1573632, 1366016, 530432, 158720, 21824925, 15302031, 2715243, 74601, 302273664, 161855232, 14872704, 2854894485, 2600075865, 712988175, 59062275

Offset: 0

Alois P. Heinz, May 22 2014

Sum_{k>0} k*T(n,k) = A249249(n).

			T(7,3) = 12 because 12 permutations of {1,2,3,4,5,6,7} have exactly 3 (possibly overlapping) occurrences of the consecutive step pattern up, up, up given by the binary expansion of 7 = 111_2: (1,2,3,4,5,7,6), (1,2,3,4,6,7,5), (1,2,3,5,6,7,4), (1,2,4,5,6,7,3), (1,3,4,5,6,7,2), (2,1,3,4,5,6,7), (2,3,4,5,6,7,1), (3,1,2,4,5,6,7), (4,1,2,3,5,6,7), (5,1,2,3,4,6,7), (6,1,2,3,4,5,7), (7,1,2,3,4,5,6).
Triangle T(n,k) begins:
: n\k :       0        1       2       3  4  ...
+-----+------------------------------------
:  0  :       1;
:  1  :       1;                             [row  1 of A008292]
:  2  :       2;                             [row  2 of A008303]
:  3  :       5,       1;                    [row  3 of A162975]
:  4  :      21,       3;                    [row  4 of A242819]
:  5  :      70,      50;                    [row  5 of A227884]
:  6  :     450,     270;                    [row  6 of A242819]
:  7  :    4326,     602,     99,     12, 1; [row  7 of A220183]
:  8  :   34944,    5376;                    [row  8 of A242820]
:  9  :  209863,  139714,  13303;            [row  9 of A230695]
: 10  : 1573632, 1366016, 530432, 158720;    [row 10 of A230797]

Alois P. Heinz, Rows n = 0..130, flattened

Column k=0-10 give: A242785, A246221, A246222, A246223, A246224, A246225, A246226, A246227, A246228, A246229, A243105.
Row sums give A000142.
Cf. A242784, A249249, A295987, A335308.

T:= proc(n) option remember; local b, k, r, h;
      k:= iquo(n,2,'r'); h:= 2^ilog2(n);
      b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand(
      add(b(u-j, o+j-1, irem(2*t,   h))*`if`(r=0 and t=k, x, 1), j=1..u)+
      add(b(u+j-1, o-j, irem(2*t+1, h))*`if`(r=1 and t=k, x, 1), j=1..o)))
      end: forget(b);
      (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 0))
    end:
seq(T(n), n=0..15);

T[n_] := T[n] = Module[{b, k, r, h}, {k, r} = QuotientRemainder[n, 2]; h = 2^Floor[Log[2, n]]; b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, Expand[ Sum[b[u - j, o + j - 1, Mod[2*t, h]]*If[r == 0 && t == k, x, 1], {j, 1, u}] + Sum[b[u + j - 1, o - j, Mod[2*t + 1, h]]*If[r == 1 && t == k, x, 1], {j, 1, o}]]]; Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0, 0]]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Feb 20 2016, after Alois P. Heinz *)

A117158 Number of permutations avoiding the consecutive pattern 1234.

Original entry on oeis.org

1, 1, 2, 6, 23, 111, 642, 4326, 33333, 288901, 2782082, 29471046, 340568843, 4263603891, 57482264322, 830335952166, 12793889924553, 209449977967081, 3630626729775362, 66429958806679686, 1279448352687538463, 25874432578888440471, 548178875969847203202

Offset: 0

Steven Finch, Apr 26 2006

nonn

a(n) is the number of permutations on [n] that avoid the consecutive pattern 1234. It is the same as the number of permutations which avoid 4321.

F. N. David and D. E. Barton, Combinatorial Chance, Hafner, New York, 1962, pages 156-157.

Alois P. Heinz, Table of n, a(n) for n = 0..450 (terms n = 0..60 from Ray Chandler)
A. Baxter, B. Nakamura, and D. Zeilberger, Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes.
Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns, Adv. Appl. Math. 36 (2006), 138-155.
Sergi Elizalde and Marc Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-125.
Steven Finch, Pattern-Avoiding Permutations. [Archived version]
Steven Finch, Pattern-Avoiding Permutations. [Cached copy, with permission]
Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory A 53 (1990), 257-285.
Ira M. Gessel, Yan Zhuang, Counting permutations by alternating descents, arXiv:1408.1886 [math.CO], 2014. See displayed equation before Eq. (3) and set m=4. - _N. J. A. Sloane_, Aug 11 2014
Kaarel Hänni, Asymptotics of descent functions, arXiv:2011.14360 [math.CO], Nov 29 2020, p. 14.
Mingjia Yang, Doron Zeilberger, Increasing Consecutive Patterns in Words, arXiv:1805.06077 [math.CO], 2018.
Mingjia Yang, An experimental walk in patterns, partitions, and words, Ph. D. Dissertation, Rutgers University (2020).

Cf. A022558, A049774, A111004, A113228, A113229, A117156, A117226, A177523, A177533, A201692, A201693, A230051, A324132.
Column k=0 of A220183.
Column k=7 of A242784.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t<2, add(b(u+j-1, o-j, t+1), j=1..o), 0)+
      add(b(u-j, o+j-1, 0), j=1..u))
    end:
a:= n-> b(n, 0, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 07 2013

a[n_]:=Coefficient[Series[2/(Cos[x]-Sin[x]+Exp[ -x]),{x,0,30}],x^n]*n!
(* second program: *)
b[u_, o_, t_] := b[u, o, t] = If[u+o==0, 1, If[t<2, Sum[b[u+j-1, o-j, t+1], {j, 1, o}], 0] + Sum[b[u-j, o+j-1, 0], {j, 1, u}]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 23 2015, after Alois P. Heinz *)

From Sergei N. Gladkovskii, Nov 30 2011: (Start)
E.g.f.: 2/(exp(-x) + cos(x) - sin(x)) = 1/W(0) with continued fraction
W(k) = 1 + (x^(2*k))/(f + f*x/(4*k + 1 - x - (4*k + 1)*b/R)), where R := x^(2*k) + b -(x^(4*k+1))/(c + (x^(2*k+1)) + x*c/T); T := 4*k + 3 - x - (4*k + 3)*d/(d +(x^(2*k+1))/W(k+1)), and
f := (4*k)!/(2*k)!; b := (4*k + 1)!/(2*k + 1)!; c := (4*k + 2)!/(2*k + 1)!; and d :=(4*k + 3)!/(2*k + 2)!. (End)
a(n) ~ n! / (sin(r)*r^(n+1)), where r = 1.0384156372665563... is the root of the equation exp(-r) + cos(r) = sin(r). - Vaclav Kotesovec, Dec 11 2013

cp's OEIS Frontend

A242783 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive step pattern given by the binary expansion of n, where 1=up and 0=down; triangle T(n,k), n>=0, read by rows.

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