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Heinz number of the n-th reversed integer partition in graded lexicographical order.
22

%I #14 Sep 22 2023 08:56:41

%S 1,2,4,3,8,6,5,16,12,10,9,7,32,24,20,18,14,15,11,64,48,40,36,28,30,22,

%T 27,21,25,13,128,96,80,72,56,60,44,54,42,50,26,45,33,35,17,256,192,

%U 160,144,112,120,88,108,84,100,52,90,66,70,34,81,63,75,39,55,49,19

%N Heinz number of the n-th reversed integer partition in graded lexicographical order.

%C A permutation of the positive integers.

%C Reversed integer partitions are finite weakly increasing sequences of positive integers. The non-reversed version is A334434.

%C This is the graded reverse of the so-called "Mathematica" order (A080577, A129129).

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

%C Also Heinz numbers of partitions in colexicographic order (cf. A211992).

%C As a triangle with row lengths A000041, the sequence starts {{1},{2},{4,3},{8,6,5},...}, 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)) = A193173(n).

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

%e 1: {} 11: {5} 44: {1,1,5}

%e 2: {1} 64: {1,1,1,1,1,1} 54: {1,2,2,2}

%e 4: {1,1} 48: {1,1,1,1,2} 42: {1,2,4}

%e 3: {2} 40: {1,1,1,3} 50: {1,3,3}

%e 8: {1,1,1} 36: {1,1,2,2} 26: {1,6}

%e 6: {1,2} 28: {1,1,4} 45: {2,2,3}

%e 5: {3} 30: {1,2,3} 33: {2,5}

%e 16: {1,1,1,1} 22: {1,5} 35: {3,4}

%e 12: {1,1,2} 27: {2,2,2} 17: {7}

%e 10: {1,3} 21: {2,4} 256: {1,1,1,1,1,1,1,1}

%e 9: {2,2} 25: {3,3} 192: {1,1,1,1,1,1,2}

%e 7: {4} 13: {6} 160: {1,1,1,1,1,3}

%e 32: {1,1,1,1,1} 128: {1,1,1,1,1,1,1} 144: {1,1,1,1,2,2}

%e 24: {1,1,1,2} 96: {1,1,1,1,1,2} 112: {1,1,1,1,4}

%e 20: {1,1,3} 80: {1,1,1,1,3} 120: {1,1,1,2,3}

%e 18: {1,2,2} 72: {1,1,1,2,2} 88: {1,1,1,5}

%e 14: {1,4} 56: {1,1,1,4} 108: {1,1,2,2,2}

%e 15: {2,3} 60: {1,1,2,3} 84: {1,1,2,4}

%e Triangle begins:

%e 1

%e 2

%e 4 3

%e 8 6 5

%e 16 12 10 9 7

%e 32 24 20 18 14 15 11

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

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

%e This corresponds to the following tetrangle:

%e 0

%e (1)

%e (11)(2)

%e (111)(12)(3)

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

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

%t lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];

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

%Y Row lengths are A000041.

%Y The constructive version is A026791 (triangle).

%Y The length-sensitive version is A185974.

%Y Compositions under the same order are A228351 (triangle).

%Y The version for non-reversed partitions is A334434.

%Y The dual version (sum/revlex) is A334436.

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

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

%Y Graded 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 order (sum/colex) are A211992.

%Y Graded Heinz numbers are given by A215366.

%Y Sorting partitions by Heinz number gives A296150.

%Y Partitions in dual Abramowitz-Stegun (sum/length/revlex) order are A334439.

%Y Cf. A056239, A066099, A129129, A228531, A333219, A333220, A334301, A334302, A334433, A334435, A334438.

%K nonn,tabf

%O 0,2

%A _Gus Wiseman_, May 03 2020