ZerosToFlagZeros from BinaryWord

The word whose reverse complement, prepended, has as many occurrences of 10 as this has zeros.

BinaryWord → BinaryWordbijectioninverse

FindStat's Mp00268, the inverse of FlagZerosToZeros.

Called as CombinatorialMap(x, "ZerosToFlagZeros"). 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

FindStat Mp00268

zeros to flag zeros

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).

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

Return a binary word of the same length, such that the number of occurrences of $10$ in the word obtained by prepending the reverse of the complement equals the number of $0$s in the original word.

Object	Image
0	0
1	1
00	10
01	00

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00268 has the rest.