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A164953
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Square array read by antidiagonals: a(m,n) = the number of different multisets (in respect to the value of the digits) of lengths of runs in the binary representations of positive integers that contain exactly m 0's and n 1's in binary. (The leftmost digit must be 1 in each binary number.)
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0
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1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 3, 4, 3, 1, 1, 4, 5, 6, 3, 1, 1, 4, 7, 8, 7, 4, 1, 1, 5, 8, 11, 10, 10, 4, 1, 1, 5, 11, 14, 15, 15, 11, 5, 1, 1, 6, 12, 20, 18, 23, 18, 14, 5, 1, 1, 6, 15, 23, 28, 28, 29, 24, 16, 6, 1, 1, 7, 17, 30, 34, 38, 37, 40, 29, 19, 6, 1, 1, 7, 20, 35, 46, 52, 51, 52, 50
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OFFSET
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0,5
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COMMENTS
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The top row of the array is where m=0. The leftmost column of the array is where n=1.
Clarification regarding the definition: Each positive integer can be thought of as a finite binary string with 1 as the leftmost digit. The "runs" alternate between those completely of 1's and those completely of 0's. Each run of digit b (0 or 1) is bounded by the digit 1-b or by the edge of the string. By "multiset of lengths (in respect to the values of the digits)" of runs, it is meant that the lengths of the runs of digit b's (b=0 or 1) form a permutation of the lengths of the runs of b's in all binary number with the same multisets of the lengths of runs. We are concerned with two multisets, those of the lengths of the runs of 0's, and those of the lengths of the runs of 1's. (See example.)
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LINKS
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EXAMPLE
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Consider those binary numbers with exactly four 1's and two 0's. There are 10 such binary numbers that each have a 1 as the leftmost digit. These binary numbers, grouped by those numbers with the same types of runs, are: (111100), (111010, 101110), (111001, 100111), (110110), (110101, 101101, 101011), (110011). There are 6 such groupings, so a(2,4) = 6.
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Definition and comment line improved by Leroy Quet, Sep 02 2009
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STATUS
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approved
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