cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A242784 Number A(n,k) of permutations of [n] avoiding the consecutive step pattern given by the binary expansion of k, where 1=up and 0=down; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 2, 5, 8, 1, 1, 1, 1, 2, 6, 17, 16, 1, 1, 1, 1, 2, 6, 21, 70, 32, 1, 1, 1, 1, 2, 6, 19, 90, 349, 64, 1, 1, 1, 1, 2, 6, 21, 70, 450, 2017, 128, 1, 1, 1, 1, 2, 6, 23, 90, 331, 2619, 13358, 256, 1, 1
Offset: 0

Views

Author

Alois P. Heinz, May 22 2014

Keywords

Examples

			A(4,5) = 19 because there are 4! = 24 permutations of {1,2,3,4} and only 5 of them do not avoid the consecutive step pattern up, down, up given by the binary expansion of 5 = 101_2: (1,3,2,4), (1,4,2,3), (2,3,1,4), (2,4,1,3), (3,4,1,2).
Square array A(n,k) begins:
  1, 1,   1,     1,     1,     1,     1,     1,     1, ...
  1, 1,   1,     1,     1,     1,     1,     1,     1, ...
  1, 1,   2,     2,     2,     2,     2,     2,     2, ...
  1, 1,   4,     5,     6,     6,     6,     6,     6, ...
  1, 1,   8,    17,    21,    19,    21,    23,    24, ...
  1, 1,  16,    70,    90,    70,    90,   111,   116, ...
  1, 1,  32,   349,   450,   331,   450,   642,   672, ...
  1, 1,  64,  2017,  2619,  1863,  2619,  4326,  4536, ...
  1, 1, 128, 13358, 17334, 11637, 17334, 33333, 34944, ...

Links

Crossrefs

Columns give: 0, 1: A000012, 2: A011782, 3: A049774, 4, 6: A177479, 5: A177477, 7: A117158, 8, 14: A177518, 9: A177519, 10: A177520, 11, 13: A177521, 12: A177522, 15: A177523, 16, 30: A177524, 17: A177525, 18, 22: A177526, 19, 25: A177527, 20, 26: A177528, 21: A177529, 23, 29: A177530, 24, 28: A177531, 27: A177532, 31: A177533, 32, 62: A177534, 33: A177535, 34, 46: A177536, 35, 49: A177537, 36, 54: A177538, 37, 41: A177539, 38: A177540, 39, 57: A177541, 40, 58: A177542, 42: A177543, 43, 53: A177544, 44, 50: A177545, 45: A177546, 47, 61: A177547, 48, 60: A177548, 51: A177549, 52: A177550, 55, 59: A177551, 56: A177552, 63: A177553, 127: A230051, 255: A230231, 511: A230232, 1023: A230233, 2047: A254523.
Main diagonal gives A242785.
Cf. A242783, A335308.

Programs

  • Maple
    A:= proc(n, k) option remember; local b, m, r, h;
      if k<2 then return 1 fi;
      m:= iquo(k, 2, 'r'); h:= 2^ilog2(k);
      b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t=m and r=0, 0, add(b(u-j, o+j-1, irem(2*t, h)), j=1..u))+
      `if`(t=m and r=1, 0, add(b(u+j-1, o-j, irem(2*t+1, h)), j=1..o)))
      end; forget(b);
      b(n, 0, 0)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..15);
  • Mathematica
    Clear[A]; A[n_, k_] := A[n, k] = Module[{b, m, r, h}, If[k < 2, Return[1]]; {m, r} = QuotientRemainder[k, 2]; h = 2^Floor[Log[2, k]]; b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, If[t == m && r == 0, 0, Sum[b[u - j, o + j - 1, Mod[2*t, h]], {j, 1, u}]] + If[t == m && r == 1, 0, Sum[b[u + j - 1, o - j, Mod[2*t + 1, h]], {j, 1, o}]]]; b[n, 0, 0]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Sep 22 2014, translated from Maple *)

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.

Original entry on oeis.org

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

Views

Author

Alois P. Heinz, May 22 2014

Keywords

Comments

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

Examples

			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]

Links

Crossrefs

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.

Programs

  • Maple
    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);
  • Mathematica
    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 *)

A335308 Number of permutations p of [n] such that the sequence of ascents and descents of p is encoded by the 0's and 1's, respectively, in the binary expansion of n (read from right to left and using leading 0's if necessary).

Original entry on oeis.org

1, 0, 0, 1, 3, 16, 26, 20, 69, 370, 1006, 945, 1266, 3015, 2365, 1001, 4367, 24736, 76960, 69615, 138397, 322944, 286824, 133056, 159391, 546504, 978054, 674245, 531530, 957320, 495495, 142506, 906191, 5537808, 18828096, 16231039, 37000909, 81351936, 71761536
Offset: 0

Views

Author

Alois P. Heinz, Sep 12 2020

Keywords

Examples

			a(0) = 1: (), the empty permutation.
a(3) = 1: 321 (down, down).
a(4) = 3: 1243, 1342, 2341 (up, up, down).
a(5) = 16: 21435, 21534, 31425, 31524, 32415, 32514, 41325, 41523, 42315, 42513, 43512, 51324, 51423, 52314, 52413, 53412 (down, up, down, up).
a(6) = 26: 143256, 153246, 154236, 163245, 164235, 165234, 243156, 253146, 254136, 263145, 264135, 265134, 342156, 352146, 354126, 362145, 364125, 365124, 452136, 453126, 462135, 463125, 465123, 562134, 563124, 564123 (up, down, down, up, up).
a(7) = 20: 4321567, 5321467, 5421367, 5431267, 6321457, 6421357, 6431257, 6521347, 6531247, 6541237, 7321456, 7421356, 7431256, 7521346, 7531246, 7541236, 7621345, 7631245, 7641235, 7651234 (down^3, up^3).

Links

Crossrefs

Cf. A060351, A242783, A242784, A242785.

Programs

  • Maple
    b:= proc(u, o, t) option remember; `if`(u+o=0, `if`(t=0, 1, 0),
     `if`(irem(t, 2)=0, add(b(u-j, o+j-1, iquo(t, 2)), j=1..u),
      add(b(u+j-1, o-j, iquo(t, 2)), j=1..o)))
    end:
a:= n-> b(n, 0, 2*n):
seq(a(n), n=0..42);

Formula

a(n) = A060351(n,n).
a(2^n-1) = binomial(2^n-2,n).
a(2^n) = binomial(2^n,n+1)-1.
Showing 1-3 of 3 results.

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