BinaryWord from IntegerPartition

The shape's boundary read from the first row down: 1 for a step down, 0 for a step left.

IntegerPartition → BinaryWord

FindStat's Mp00095. Injective.

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

Live

Explore

Each element of IntegerPartition at a size, and its image, applied in your browser; sizes past 7 are not offered.

Size

Properties

Sizes 1 to

Transport a statistic

Pick a statistic on BinaryWord. Its distribution over the images of IntegerPartition at each size is computed, and compared with the 165 statistics on IntegerPartition: 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: DyckPath, Finset.

FindStat Mp00095

to binary word

Matched by value: our map agrees with FindStat's image on each of the 29 rows of size at most 6 we can read (0 unread; its table has 1200 rows).

Injectionas FindStat records them, checked by FindStat's contributors.

Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.

Object	Image
[1]	10
[2]	100
[3]	1000
[4]	10000

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00095 has the rest.