A290257 a(n) is the size of the Durfee square of the integer partition having viabin number n.

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 2, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 2, 2, 1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 2, 3, 2, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 2, 2, 1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 2, 3, 2, 2, 1, 2, 2, 2, 2, 3

Offset

1,6

Comments

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 [2,2,2,1]. The southeast border of its Ferrers board yields 10100, leading to the viabin number 20.

References

G. E. Andrews and K. Ericksson, Integer Partitions, Cambridge University Press 2004.

Links

Table of n, a(n) for n=1..100.

Formula

a(n)= largest k such that the first k 1's in the binary representation of 2n precedes the last k 0's; a(n)=1 if there are no zeros. [Clarified by George Beck, Apr 30 2021]

Example

a(20) = 2. Indeed, the integer partition corresponding to the viabin number 20 is [2,2,2,1] (see 1st comment); the size of its Durfee square is 2.

Maple

with(ListTools): b := proc (n) local L, X1, X2, k, u: L := Reverse(convert(n, base, 2)): X1 := [SearchAll(1, L)]: X2 := Reverse([SearchAll(0, L)]): u := 0: for k to min(nops(X1), nops(X2)) do if X1[k] < X2[k] then u := k else break end if end do: u end proc: a := proc (n) options operator, arrow: b(2*n) end proc: seq(a(n), n = 1 .. 100); # due to W. Edwin Clark

Mathematica

A290257seq@n_ := Module[
   {x, p1, p2, m, p1new, p2new, k},
   x = IntegerDigits[2 n, 2];
   p1 = First /@ Position[x, 1];
   p2 = First /@ Position[x, 0];
   m = Min[Length /@ {p1, p2}];
   p1new = Take[p1, m];
   p2new = Reverse@Take[p2, -m];
   k = Position[Positive[p2new - p1new], True];
   If[k == {}, 1, First@Last@k]
   ];
A290257seq /@ Range@100
~~

Crossrefs

Sequence in context: A366643 A212185 A270928 * A196942 A184304 A363298

Adjacent sequences: A290254 A290255 A290256 * A290258 A290259 A290260

Keyword

nonn

Author

Emeric Deutsch, Sep 05 2017

Status

approved