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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A152884 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} with excedance set equal to the k-th subset of {1,2,...,n-1} (n>=0, 0<=k<=ceiling(2^(n-1))-1). The subsets of {1,2,...,n-1} are ordered according to size, while the subsets of the same size follow the lexicographic order.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 3, 1, 7, 3, 1, 1, 1, 15, 7, 3, 1, 31, 17, 7, 7, 3, 1, 15, 7, 3, 1, 1, 1, 31, 15, 7, 3, 1, 115, 69, 37, 15, 31, 17, 7, 7, 3, 1, 115, 69, 31, 37, 17, 7, 15, 7, 3, 1, 31, 15, 7, 3, 1, 1, 1, 63, 31, 15, 7, 3, 1, 391, 245, 145, 77, 31, 115, 69, 37, 15, 31, 17, 7, 7, 3, 1
Offset: 0

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Author

Emeric Deutsch, Jan 13 2009

Keywords

Comments

For example, the eight subsets of {1,2,3} are ordered as empty,1,2,3,12,13,23,123. The excedance set of a permutation p of {1,2,...,n} is the set of indices i such that p(i)>i; it is a subset of {1,2,...,n-1}.
Row n contains ceiling(2^(n-1)) entries.
Sum of entries in row n is n! (A000142).
The given Maple program yields the term of the sequence corresponding to a specified permutation size n and a specified excedance set A.
All terms are odd. - Alois P. Heinz, Jan 31 2023

Examples

			T(5,3) = 3 because the 3rd subset of {1,2,3,4} is {3} and the permutations of {1,2,3,4,5} with excedance set {3} are 12435, 12534 and 12543.
T(5,4) = 1: 12354 (4th subset of {1,2,3,4} is {4}).
Triangle starts:
      k=0   1  2  3  4   5   6  7  8 ...
  n=0:  1;
  n=1:  1;
  n=2:  1,  1;
  n=3:  1,  3, 1, 1;
  n=4:  1,  7, 3, 1, 7,  3,  1, 1;
  n=5:  1, 15, 7, 3, 1, 31, 17, 7, 7, 3, 1, 15, 7, 3, 1, 1;
  ...

Links

Crossrefs

Row sums are A000142.
See A360288, A360289 for similar triangles.
Cf. A000225, A011782, A082185, A136126, A193360, A329369 (another version).

Programs

  • Maple
    n := 7: A := {1, 2, 4}: with(combinat): P := permute(n): EX := proc (p) local S, i: S := {}: for i to n-1 do if i < p[i] then S := `union`(S, {i}) else end if end do: S end proc: ct := 0: for j to factorial(n) do if EX(P[j]) = A then ct := ct+1 else end if end do: ct;
# second Maple program:
T:= proc(n) option remember; uses combinat; local b, i, l;
      l:= map(x-> {x[]}, [seq(choose([$1..n-1], i)[], i=0..n-1)]):
      for i to nops(l) do h(l[i]):=i od:
      b:= proc(s, l) option remember; (m->
           `if`(m=0, x^h(l), add(b(s minus {i}, {l[],
           `if`(i
      seq(coeff(p, x, i), i=1..degree(p)))(b({$1..n}, {}))
    end: T(0):=1:
seq(T(n), n=0..8);  # Alois P. Heinz, Jan 29 2023

Formula

T(n,k) = A000225(n-k) = 2^(n-k) - 1 for n>k>0. - Alexander R. Povolotsky, May 14 2025

Extensions

T(0,0)=1 prepended and indexing adapted by Alois P. Heinz, Jan 29 2023

A282903 Concatenation of elements of P{d_1}, P{d_2}, P{d_3}, ..., P{d_n}; where P{d_n} denote the power set of the set of divisors of n ordered by the size of the subsets, and in each subset, following the increasing order.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 4, 1, 2, 1, 4, 2, 4, 1, 2, 4, 1, 5, 1, 5, 1, 2, 3, 6, 1, 2, 1, 3, 1, 6, 2, 3, 2, 6, 3, 6, 1, 2, 3, 1, 2, 6, 1, 3, 6, 2, 3, 6, 1, 2, 3, 6, 1, 7, 1, 7, 1, 2, 4, 8, 1, 2, 1, 4, 1, 8, 2, 4, 2, 8, 4, 8, 1, 2, 4, 1, 2, 8, 1, 4, 8, 2
Offset: 1

Views

Author

Jaroslav Krizek, Feb 24 2017

Keywords

Examples

			Rows with power sets of divisors of n (without nonempty sets):
{1};
{1}, {2}, {1, 2};
{1}, {3}, {1, 3};
{1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4};
{1}, {5}, {1, 5};
...
Concatenation: 1, 1, 2, 1, 2, 1, 3, 1, 3, ...

Crossrefs

Cf. A193360 (concatenation of P{1..n}).

A360302 T(n,k) is the position of the set encoded in the binary expansion of k within the shortlex order for the powerset of [n]; triangle T(n,k), n>=0, 0<=k<=2^n-1, read by rows.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 3, 0, 1, 2, 4, 3, 5, 6, 7, 0, 1, 2, 5, 3, 6, 8, 11, 4, 7, 9, 12, 10, 13, 14, 15, 0, 1, 2, 6, 3, 7, 10, 16, 4, 8, 11, 17, 13, 19, 22, 26, 5, 9, 12, 18, 14, 20, 23, 27, 15, 21, 24, 28, 25, 29, 30, 31, 0, 1, 2, 7, 3, 8, 12, 22, 4, 9, 13, 23, 16
Offset: 0

Views

Author

Alois P. Heinz, Feb 03 2023

Keywords

Comments

In shortlex order for 2^[n] the subsets are primarily sorted by cardinality and then into lexicographical order.
The set encoded by k consists of the indices of 1-bits (rightmost index is 1).
Row n is a permutation of {0, 1, ..., 2^n-1} whose inverse is in row n of A359941.

Examples

			The subsets of [4] listed in shortlex order (starting at position 0) are: {}, {1}, {2}, {3}, {4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4}, {1,2,3,4}.
T(4,0) = T(4,0000_2) = 0: {} is at position 0.
T(4,3) = T(4,0011_2) = 5: {1,2} is at position 5.
T(4,6) = T(4,0110_2) = 8: {2,3} is at position 8.
T(4,7) = T(4,0111_2) = 11: {1,2,3} is at position 11.
T(4,15) = T(4,1111_2) = 15: {1,2,3,4} is at position 15.
Triangle T(n,k) begins:
  0;
  0, 1;
  0, 1, 2, 3;
  0, 1, 2, 4, 3, 5, 6,  7;
  0, 1, 2, 5, 3, 6, 8, 11, 4, 7, 9, 12, 10, 13, 14, 15;
  ...

Links

Crossrefs

Columns k=0-1 give: A000004, A057427.
Row sums give A006516(n) = A000217(A000225(n)).
Row lengths are A000079.
Cf. A082185, A193360, A359941.

Programs

  • Maple
    T:= proc(n) option remember; local h, i, l;
      l:= map(x-> add(2^(i-1), i=x),
         [seq(combinat[choose]([$1..n], i)[], i=0..n)]);
      h(0):=0; for i to nops(l) do h(l[i]):= (i-1) od:
      seq(h(i), i=0..2^n-1)
    end:
seq(T(n), n=0..6);

Formula

T(n,A359941(n,k)) = k = A359941(n,T(n,k)).

A282904 Concatenation of the numbers of elements of P{1}, P{1, 2}, P{1, 2, 3}, ..., P{1, 2, 3, ..., n}; where P{A} denote the power set of set A ordered by the size of the subsets, and in each subset, following the increasing order.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 3, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3
Offset: 1

Views

Author

Jaroslav Krizek, Feb 24 2017

Keywords

Examples

			Rows with power sets of set of numbers from 1 to n (without nonempty sets):
{1};
{1}, {2}, {1, 2};
{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3};
...
Rows with the number of elements of these subsets:
1;
1, 1, 2;
1, 1, 1, 2, 2, 2, 3;
...
Concatenation: 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 3, ...

Crossrefs

Cf. A193360, A282905, A221914.

A282905 Concatenation of the sums of elements of P{1}, P{1, 2}, P{1, 2, 3}, ..., P{1, 2, 3, ..., n}; where P{A} denote the power set of set A ordered by the size of the subsets, and in each subset, following the increasing order.

Original entry on oeis.org

1, 1, 2, 3, 1, 2, 3, 3, 4, 5, 6, 1, 2, 3, 4, 3, 4, 5, 5, 6, 7, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 3, 4, 5, 6, 5, 6, 7, 7, 8, 9, 6, 7, 8, 8, 9, 10, 9, 10, 11, 12, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, 6, 3, 4, 5, 6, 7, 5, 6, 7, 8, 7, 8, 9, 9, 10, 11, 6, 7, 8, 9, 8
Offset: 1

Views

Author

Jaroslav Krizek, Feb 24 2017

Keywords

Examples

			Rows with power sets of set of numbers from 1 to n (without nonempty sets):
{1};
{1}, {2}, {1, 2};
{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3};
...
Rows with the sum of elements of these subsets:
1;
1, 2, 3;
1, 2, 3, 3, 4, 5, 6;
...
Concatenation: 1, 1, 2, 3, 1, 2, 3, 3, 4, 5, 6, ...

Crossrefs

Cf. A193360, A282904, A221914.
Showing 1-5 of 5 results.

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