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A295261
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Partitions into parts with frequency less than or equal to their place in the list of summands.
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2
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1, 1, 1, 2, 2, 4, 4, 6, 8, 11, 12, 18, 22, 28, 34, 44, 54, 69, 82, 102, 125, 154, 185, 226, 271, 327, 393, 474, 562, 673, 797, 947, 1124, 1329, 1563, 1846, 2164, 2541, 2974, 3480, 4062, 4738, 5508, 6403, 7432, 8614, 9966, 11530, 13307, 15345, 17670, 20337
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OFFSET
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0,4
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COMMENTS
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Let the summands of a partition be s(1) < s(2) < ... < s(k) and the frequency of s(i) be f(i). Then we count those partitions for which f(i) <= i.
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LINKS
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EXAMPLE
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The partition 1+1 is not counted because its smallest part, 1, appears twice.
The partition 3+2+2+1 is counted because its smallest part, 1, appears once; its next smallest part, 2 appears twice (and 2 <= 2) and its third part, 3, appears 1 time (and 1 <= 3).
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MAPLE
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b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(n<i, 0,
add(b(n-i*j, i+1, t+`if`(j=0, 0, 1)), j=0..min(t, n/i))))
end:
a:= n-> b(n, 1$2):
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MATHEMATICA
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<< Combinatorica`;
nend = 30;
For[n = 1, n <= nend, n++, count[n] = 0;
part = Partitions[n];
For[i = 1, i <= Length[part], i++,
t = Tally[part[[i]]];
condition = True;
For[j = 1, j <= Length[t], j++,
If[t[[-j, 2]] > j, condition = False ]];
If[condition, count[n]++]]];
Print[Table[count[i], {i, 1, nend}]]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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