

A161919


Permutation of natural numbers: concatenation of subsequences A161924(A000070(k1)..A026905(k)), k >= 1, each sorted into ascending order.


5



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
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OFFSET

1,2


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


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


EXAMPLE

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.


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, i1)[],
`if`(i>n, [], map(x> [x[], i], b(ni, 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[(n1)/2] + 1];
Clear[T]; T[_, _] = 0; Do[T[row[k], col[k]] = k, {k, 1, 2^columns}];
Table[DeleteCases[Sort @ Table[T[n1, k], {n, 1, 2^(k1)}], 0], {k, 1, columns}] // Flatten (* JeanFrançois Alcover, Feb 16 2021 *)


CROSSREFS

Inverse: A166277. Sequence A161924 gives the same rows before sorting.
Sequence in context: A233280 A233279 A267111 * A269391 A095903 A267112
Adjacent sequences: A161916 A161917 A161918 * A161920 A161921 A161922


KEYWORD

nonn,tabf


AUTHOR

Alford Arnold, Jun 23 2009


EXTENSIONS

Edited and extended by Antti Karttunen, Oct 12 2009


STATUS

approved



