A334435 - OEIS

A334435

Heinz numbers of all reversed integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.

%I #20 Sep 22 2023 07:53:08

%S 1,2,3,4,5,6,8,7,9,10,12,16,11,15,14,18,20,24,32,13,25,21,22,27,30,28,

%T 36,40,48,64,17,35,33,26,45,50,42,44,54,60,56,72,80,96,128,19,49,55,

%U 39,34,75,63,70,66,52,81,90,100,84,88,108,120,112,144,160,192,256

%N Heinz numbers of all reversed integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.

%C First differs from A334433 at a(75) = 99, A334433(75) = 98.

%C First differs from A334436 at a(22) = 22, A334436(22) = 27.

%C A permutation of the positive integers.

%C Reversed integer partitions are finite weakly increasing sequences of positive integers.

%C This is the Abramowitz-Stegun ordering of reversed partitions (A185974) except that the finer order is reverse-lexicographic instead of lexicographic. The version for non-reversed partitions is A334438.

%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

%C As a triangle with row lengths A000041, the sequence starts {{1},{2},{3,4},{5,6,8},...}, so offset is 0.

%H Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>

%F A001222(a(n)) = A036043(n).

%e The sequence of terms together with their prime indices begins:

%e 1: {} 32: {1,1,1,1,1} 42: {1,2,4}

%e 2: {1} 13: {6} 44: {1,1,5}

%e 3: {2} 25: {3,3} 54: {1,2,2,2}

%e 4: {1,1} 21: {2,4} 60: {1,1,2,3}

%e 5: {3} 22: {1,5} 56: {1,1,1,4}

%e 6: {1,2} 27: {2,2,2} 72: {1,1,1,2,2}

%e 8: {1,1,1} 30: {1,2,3} 80: {1,1,1,1,3}

%e 7: {4} 28: {1,1,4} 96: {1,1,1,1,1,2}

%e 9: {2,2} 36: {1,1,2,2} 128: {1,1,1,1,1,1,1}

%e 10: {1,3} 40: {1,1,1,3} 19: {8}

%e 12: {1,1,2} 48: {1,1,1,1,2} 49: {4,4}

%e 16: {1,1,1,1} 64: {1,1,1,1,1,1} 55: {3,5}

%e 11: {5} 17: {7} 39: {2,6}

%e 15: {2,3} 35: {3,4} 34: {1,7}

%e 14: {1,4} 33: {2,5} 75: {2,3,3}

%e 18: {1,2,2} 26: {1,6} 63: {2,2,4}

%e 20: {1,1,3} 45: {2,2,3} 70: {1,3,4}

%e 24: {1,1,1,2} 50: {1,3,3} 66: {1,2,5}

%e Triangle begins:

%e 1

%e 2

%e 3 4

%e 5 6 8

%e 7 9 10 12 16

%e 11 15 14 18 20 24 32

%e 13 25 21 22 27 30 28 36 40 48 64

%e 17 35 33 26 45 50 42 44 54 60 56 72 80 96 128

%e This corresponds to the following tetrangle:

%e 0

%e (1)

%e (2)(11)

%e (3)(12)(111)

%e (4)(22)(13)(112)(1111)

%e (5)(23)(14)(122)(113)(1112)(11111)

%t revlensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]<Length[c],OrderedQ[{c,f}]];

%t Table[Times@@Prime/@#&/@Sort[Sort/@IntegerPartitions[n],revlensort],{n,0,8}]

%Y Row lengths are A000041.

%Y The dual version (sum/length/lex) is A185974.

%Y Compositions under the same order are A296774 (triangle).

%Y The constructive version is A334302.

%Y Ignoring length gives A334436.

%Y The version for non-reversed partitions is A334438.

%Y Partitions in this order (sum/length/revlex) are A334439.

%Y Lexicographically ordered reversed partitions are A026791.

%Y Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.

%Y Partitions in increasing-length colex order (sum/length/colex) are A036037.

%Y Reverse-lexicographically ordered partitions are A080577.

%Y Sorting reversed partitions by Heinz number gives A112798.

%Y Graded lexicographically ordered partitions are A193073.

%Y Partitions in colexicographic (sum/colex) order are A211992.

%Y Graded Heinz numbers are given by A215366.

%Y Sorting partitions by Heinz number gives A296150.

%Y Cf. A056239, A124734, A129129, A228100, A228531, A333219, A333220, A334301, A334433, A334434, A334437.

%K nonn,tabf

%O 0,2

%A _Gus Wiseman_, May 02 2020