A299774 Irregular triangle read by rows in which row n lists the indices of the partitions into equal parts in the list of colexicographically ordered partitions of n.
1, 1, 2, 1, 3, 1, 4, 5, 1, 7, 1, 8, 10, 11, 1, 15, 1, 16, 21, 22, 1, 27, 30, 1, 31, 41, 42, 1, 56, 1, 57, 69, 73, 76, 77, 1, 101, 1, 102, 134, 135, 1, 160, 172, 176, 1, 177, 221, 230, 231, 1, 297, 1, 298, 353, 380, 384, 385, 1, 490, 1, 491, 604, 615, 626, 627, 1
Offset: 1
Examples
Triangle begins: 1; 1, 2; 1, 3; 1, 4, 5; 1, 7; 1, 8, 10, 11; 1, 15; 1, 16, 21, 22; 1, 27, 30; 1, 31, 41, 42; 1, 56; 1, 57, 69, 73, 76, 77; 1, 101; 1, 102, 134, 135; 1, 160, 172, 176; ... For n = 6 the partitions of 6 into equal parts are [1, 1, 1, 1, 1, 1], [2, 2, 2], [3, 3] and [6]. Then we have that in the list of colexicographically ordered partitions of 6 these partitions are in the rows 1, 8, 10 and 11 respectively as shown below, so the 6th row of the triangle is [1, 8, 10, 11]. ------------------------------------------------------------- p Diagram Partitions of 6 ------------------------------------------------------------- _ _ _ _ _ _ 1 |_| | | | | | [1, 1, 1, 1, 1, 1] <--- equal parts 2 |_ _| | | | | [2, 1, 1, 1, 1] 3 |_ _ _| | | | [3, 1, 1, 1] 4 |_ _| | | | [2, 2, 1, 1] 5 |_ _ _ _| | | [4, 1, 1] 6 |_ _ _| | | [3, 2, 1] 7 |_ _ _ _ _| | [5, 1] 8 |_ _| | | [2, 2, 2] <--- equal parts 9 |_ _ _ _| | [4, 2] 10 |_ _ _| | [3, 3] <--- equal parts 11 |_ _ _ _ _ _| [6] <--- equal parts .
Crossrefs
Programs
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PARI
row(n) = {if(n == 1, return([1])); my(nd = numdiv(n), res = vector(nd)); res[1] = 1; res[nd] = numbpart(n); if(nd > 2, t = nd - 1; p = vecsort(partitions(n)); forstep(i = #p - 1, 2, -1, if(p[i][1] == p[i][#p[i]], res[t] = i; t--; if(t==1, return(res)))), return(res))} \\ David A. Corneth, Aug 17 2018
Extensions
Terms a(46) and beyond from David A. Corneth, Aug 16 2018
Comments