# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a290253 Showing 1-1 of 1 %I A290253 #21 Aug 06 2024 09:55:30 %S A290253 0,1,1,1,2,1,1,1,2,1,2,2,3,1,1,1,1,2,1,1,2,2,1,3,1,2,2,2,3,2,3,3,4,1, %T A290253 1,1,1,1,2,1,1,1,2,2,1,1,3,1,1,2,2,2,1,3,2,1,3,3,1,4,1,2,2,2,2,3,2,2, %U A290253 3,3,2,4,2,3,3,3,4,3,4,4,5,1,1,1,1,1,1,2,1,1,1,1,2,2,1,1,1,3,1,1,1,2,2,2,1,1,3,2,1,1,3,3,1,1,4,1,1,2,2,2,2,1,3,2,2,1,3,3,2,1,4,2,1,3,3,3,1,4,3,1,4,4,1,5,1,2,2,2,2,2,3,2,2,2,3,3,2,2 %N A290253 Triangle read by rows. Row n consists of the parts, ordered nonincreasingly, of the integer partition having viabin number n. %C A290253 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. %C A290253 Number of entries in row n is A290251(n). %C A290253 In the Maple program the command vitopart(n) yields the integer partition having viabin number n. %H A290253 Alois P. Heinz, Rows n = 0..4095, flattened %e A290253 Row 25 is 3,2,2. Indeed, the binary form of 25 is 11001. Consequently, the southeast border of the Ferrers board of the associated partition is EENNEN, where E and N are the steps [1,0] and [0,1], respectively. This leads to the partition [3,2,2]. %e A290253 Triangle begins: %e A290253 0, %e A290253 1; %e A290253 1,1; %e A290253 2; %e A290253 1,1,1; %e A290253 2,1; %e A290253 2,2; %e A290253 3; %p A290253 # (due to _W. Edwin Clark_) %p A290253 vitopart := proc (n) local L, i, j, N, p, t; N := 2*n; L := ListTools:-Reverse(convert(N, base, 2)); j := 0; for i to nops(L) do if L[i] = 0 then j := j+1; p[j] := numboccur(L[1 .. i], 1) end if end do; sort([seq(p[t], t = 1 .. j)], `>=`) end proc: %p A290253 # second Maple program: %p A290253 T:= proc(n) local m; m:= n; [0]; while m>0 do `if`(1= %p A290253 irem(m, 2, 'm'), map(x-> x+1, %), [%[], 0]) od: %[] %p A290253 end: %p A290253 seq(T(n), n=0..50); # _Alois P. Heinz_, Aug 23 2017 %t A290253 T[n_] := Module[{L = IntegerDigits[2n, 2], j = 0, p}, Do[If[L[[i]] == 0, j++; p[j] = Count[L[[;;i]], 1]], {i, 1, Length[L]}]; Array[p, j] // Reverse]; %t A290253 Table[T[n], {n, 0, 50}] // Flatten (* _Jean-François Alcover_, Aug 06 2024, after _W. Edwin Clark_ *) %Y A290253 Row sums give A161511. %Y A290253 Row lengths give A008687(n+1). %Y A290253 Cf. A253563, A290251. %K A290253 nonn,tabf %O A290253 0,5 %A A290253 _Emeric Deutsch_, Aug 23 2017 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE