RunSort from BinaryWord

The word with its weakly increasing runs sorted lexicographically.

BinaryWord → BinaryWordidempotent

FindStat's Mp00224. Lossy: the runs' order is forgotten. The runs are the maximal stretches 0...01...1.

Called as CombinatorialMap(x, "RunSort"). The head's page has its examples.

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Each element of BinaryWord at a size, and its image, applied in your browser; sizes past 6 are not offered.

Size

Properties

Sizes 1 to

Transport a statistic

Pick a statistic on BinaryWord. Its distribution over the images of BinaryWord at each size is computed, and compared with the 59 statistics on BinaryWord: those that share it are the ones the map sends it to, as far as the sizes computed show.

Statistic HammingWeightLeadingOnesCountOnePositionSumOneRunCountOddOneRunCountLongestRunOfOnesChangeCountPathDepthOnesMinusZerosBalancedPrefixCountSymmetryDegreeAscentsInversionsMajorIndexLongestConstantRunLongestAlternatingFactorFirstOnePositionAfterAppendedOneDecimalRepresentationDecimalRepresentationWithLeadingOneDistinctFactorCountBorderLengthSymmetricBorderLengthInversionSumNonInversionSumStrictlyIncreasingRunsMinimalPeriodBorderCountWeakMajorIndexDistinctSubsequenceCountRepetitionExponentBalanceShortestPalindromicDecompositionLengthPalindromicDecompositionCountLongestCatalanFactorSemilengthCatalanFactorCountCriticalStepCountInternalInversionsUniqueFactorLengthLyndonFactorizationLengthLongestYamanouchiPrefixLongestPalindromicPrefixLongestPalindromicFactorLongestPalindromicSubsequenceDistinctCubeCountDistinctSquareCountCyclicShiftIndexDescentWordPreimageCountLatticePathsAboveWordCountSameEndpointPathsAboveWordCountEvenPalindromicPrefixHalfLengthOddPalindromicPrefixHalfLengthLongestPalindromicFactorStartingWithOneReverseComplementFactorHalfLengthReverseComplementFactorHalfLengthStartingWithOneFlexModularMajorIndexValleyFillingDegreeBaselineCrossingsPrimitiveFactorCount Sizes 1 to

Also from

The same name out of other carriers, each its own implementation: Permutation.

FindStat Mp00224

runsort

Matched by value: our map agrees with FindStat's image on each of the 126 rows of size at most 6 we can read (0 unread; its table has 1200 rows).

IdempotentGraded mapas FindStat records them, checked by FindStat's contributors.

The word obtained by sorting the weakly increasing runs lexicographically.

Object	Image
0	0
1	1
00	00
01	01

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00224 has the rest.