Revision History for A359043
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing all changes.
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#9 by N. J. A. Sloane at Wed Dec 21 20:11:44 EST 2022
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#8 by Gus Wiseman at Wed Dec 21 19:50:55 EST 2022
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#7 by Gus Wiseman at Wed Dec 21 19:50:49 EST 2022
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| EXAMPLE
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The 29-th29th composition in standard order is (1,1,2,1), with adjusted partial sums (1,1,2,2), with sum 6, so a(29) = 6.
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| STATUS
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proposed
editing
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#6 by Gus Wiseman at Wed Dec 21 19:49:57 EST 2022
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#5 by Gus Wiseman at Wed Dec 21 19:49:37 EST 2022
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#4 by Gus Wiseman at Wed Dec 21 19:48:48 EST 2022
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#3 by Gus Wiseman at Wed Dec 21 19:41:33 EST 2022
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| MATHEMATICA
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stc[n_]:=Differences[Prepend[Join@@ @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
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| CROSSREFS
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The unadjusted reverse version is A029931, row sums of A048793 (Heinz numbers A019565).
Row sums of A242628, ranked by A253565.
A048793 gives partial sums of reversed standard comps, Heinz number A019565.
A242628 lists adjusted partial sums of standard compositions, ranked by A253565.
A351014 counts distinct runs in standard compositions.
`Cf. A000120, A000720, A001511, A029931, A059893, A061395, A241916, A242628, A253566, A355536, A358133, A358137, A358170.
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#2 by Gus Wiseman at Wed Dec 21 18:48:25 EST 2022
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| NAME
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allocatedSum of adjusted partial sums of the n-th composition in standard order (A066099). Row forsums Gusof WisemanA242628.
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| DATA
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0, 1, 2, 2, 3, 4, 3, 3, 4, 6, 5, 6, 4, 5, 4, 4, 5, 8, 7, 9, 6, 8, 7, 8, 5, 7, 6, 7, 5, 6, 5, 5, 6, 10, 9, 12, 8, 11, 10, 12, 7, 10, 9, 11, 8, 10, 9, 10, 6, 9, 8, 10, 7, 9, 8, 9, 6, 8, 7, 8, 6, 7, 6, 6, 7, 12, 11, 15, 10, 14, 13, 16, 9, 13, 12, 15, 11, 14, 13
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| OFFSET
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0,3
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| COMMENTS
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We define the adjusted partial sums of a composition to be obtained by subtracting one from all parts, taking partial sums, and adding one back to all parts.
The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
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| LINKS
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Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>
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| EXAMPLE
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The 29-th composition in standard order is (1,1,2,1), with adjusted partial sums (1,1,2,2), with sum 6, so a(29) = 6.
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| MATHEMATICA
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stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Total[Accumulate[stc[n]-1]+1], {n, 0, 100}]
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| CROSSREFS
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See link for sequences related to standard compositions.
The reverse version is A161511, row sums of A125106 (Heinz numbers A005940).
The unadjusted version is A359042, row sums of A358134.
A011782 counts compositions.
A048793 gives partial sums of reversed standard comps, Heinz number A019565.
A066099 lists standard compositions.
A242628 lists adjusted partial sums of standard compositions, ranked by A253565.
A351014 counts distinct runs in standard compositions.
A358135 gives last minus first of standard compositions.
A358194 counts partitions by sum and weighted sum.
`Cf. A000120, A000720, A001511, A029931, A059893, A061395, A241916, A242628, A253566, A355536, A358133, A358137, A358170.
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| KEYWORD
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allocated
nonn
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| AUTHOR
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Gus Wiseman, Dec 21 2022
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| STATUS
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approved
editing
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#1 by Gus Wiseman at Tue Dec 13 16:42:22 EST 2022
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| NAME
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allocated for Gus Wiseman
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| KEYWORD
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allocated
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| STATUS
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approved
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