A161919 Permutation of natural numbers: concatenation of subsequences A161924(A000070(k-1)..A026905(k)), k >= 1, each sorted into ascending order.

1, 2, 3, 4, 5, 7, 6, 8, 9, 11, 15, 10, 13, 16, 17, 19, 23, 31, 12, 14, 18, 21, 27, 32, 33, 35, 39, 47, 63, 20, 22, 25, 29, 34, 37, 43, 55, 64, 65, 67, 71, 79, 95, 127, 24, 26, 30, 36, 38, 41, 45, 51, 59, 66, 69, 75, 87, 111, 128, 129, 131, 135, 143, 159, 191, 255, 28, 40

Offset: 1

Author

Alford Arnold, Jun 23 2009

Keywords

• nonn
• tabf

Comments

This is the lexicographically earliest sequence a_n for which it holds that A161511(a(n)) = A036042(n) for all n.

Triangle T(n,k) read by rows. Row n lists in increasing order the viabin numbers of the integer partitions of n (n >= 1, k >= 1). The viabin number of an integer partition is defined in the following way. Consider the southeast border of the Ferrers board of the integer partition and consider the binary number obtained by replacing each east step with 1 and each north step, except the last one, with 0. The corresponding decimal form is, by definition, the viabin number of the given integer partition. "Viabin" is coined from "via binary". For example, consider the integer partition [3,1,1] of 5. The southeast border of its Ferrers board yields 10011, leading to the viabin number 19 (an entry in the 5th row). - Emeric Deutsch, Sep 06 2017

After specifying the value of n, the first Maple program yields the entries of row n. - Emeric Deutsch, Feb 26 2016

After specifying the value of m, the third Maple program yields the first m rows; the command partovi(p) yields the viabin number of the partition p = [a,b,c,...]. - Emeric Deutsch, Aug 31 2017

Examples

			This can be viewed as an irregular table, where row r (>= 1) has A000041(r) elements, i.e., as 1; 2,3; 4,5,7; 6,8,9,11,15; 10,13,16,17,19,23,31; etc. A125106 illustrates how each number is mapped to a partition.
		

Links

• Alois P. Heinz, Rows n = 1..28 (first 18 rows from A. Karttunen)
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Inverse: A166277. Sequence A161924 gives the same rows before sorting.

Programs

Maple

n := 11: s := proc (b) local t, i, j: t := 0: for i to nops(b) do for j from i+1 to nops(b) do if b[j]-b[i] = 1 then t := t+1 else  end if end do end do: t end proc: A[n] := {}: for i to 2^n do a[i] := convert(2*i, base, 2) end do: for k to 2^n do if s(a[k]) = n then A[n] := `union`(A[n], {k}) else  end if end do: A[n]; # Emeric Deutsch, Feb 26 2016
# second Maple program:
f:= proc(l) local i, r; r:= 0; for i to nops(l)-1 do
       r:= 2*((x-> 2*x+1)@@(l[i+1]-l[i]))(r) od; r/2
    end:
b:= proc(n, i) `if`(n=0 or i=1, [[0, 1$n]], [b(n, i-1)[],
      `if`(i>n, [], map(x-> [x[], i], b(n-i, i)))[]])
    end:
T:= n-> sort(map(f, b(n$2)))[]:
seq(T(n), n=1..10);  # Alois P. Heinz, Jul 25 2017
# 3rd Maple program:
m := 10; with(combinat): ff := proc (X) local s: s := [1, seq(0, j = 1 .. X[2])]: s := map(convert, s, string): return cat(op(s)) end proc: partovi := proc (P) local X, n, Y, i: X := convert(P, multiset): n := X[-1][1]: Y := map(proc (t) options operator, arrow: t[1] end proc, X): for i to n do if member(i, Y) = false then X := [op(X), [i, 0]] end if end do: X := sort(X, proc (s, t) options operator, arrow: evalb(s[1] < t[1]) end proc): X := map(ff, X): X := cat(op(X)): n := parse(X): n := convert(n, decimal, binary): (1/2)*n end proc: for n to m do {seq(partovi(partition(n)[q]), q = 1 .. numbpart(n))} end do; # Emeric Deutsch, Aug 31 2017

Mathematica

columns = 10;
row[n_] := n - 2^Floor[Log2[n]];
col[0] = 0; col[n_] := If[EvenQ[n], col[n/2] + DigitCount[n/2, 2, 1], col[(n-1)/2] + 1];
Clear[T]; T[, ] = 0; Do[T[row[k], col[k]] = k, {k, 1, 2^columns}];
Table[DeleteCases[Sort @ Table[T[n-1, k], {n, 1, 2^(k-1)}], 0], {k, 1, columns}] // Flatten (* Jean-François Alcover, Feb 16 2021 *)

Extensions

Edited and extended by Antti Karttunen, Oct 12 2009