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A064428
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Number of partitions of n with nonnegative crank.
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43
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1, 0, 1, 2, 3, 4, 6, 8, 12, 16, 23, 30, 42, 54, 73, 94, 124, 158, 206, 260, 334, 420, 532, 664, 835, 1034, 1288, 1588, 1962, 2404, 2953, 3598, 4392, 5328, 6466, 7808, 9432, 11338, 13632, 16326, 19544, 23316, 27806, 33054, 39273, 46534, 55096, 65076, 76808
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OFFSET
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0,4
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COMMENTS
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For a partition p, let l(p) = largest part of p, w(p) = number of 1's in p, m(p) = number of parts of p larger than w(p). The crank of p is given by l(p) if w(p) = 0, otherwise m(p)-w(p).
Also the number of even-length compositions of n with alternating parts strictly decreasing, or properly 2-colored partitions (proper = no equal parts of the same color) with the same number of parts of each color, or ordered pairs of strict partitions of the same length with total n. The odd-length case is A001522, and there are a total of A000041 compositions with alternating parts strictly decreasing (see A342528 for a bijective proof). The a(2) = 1 through a(7) = 8 ordered pairs of strict partitions of the same length are:
(1)(1) (1)(2) (1)(3) (1)(4) (1)(5) (1)(6)
(2)(1) (2)(2) (2)(3) (2)(4) (2)(5)
(3)(1) (3)(2) (3)(3) (3)(4)
(4)(1) (4)(2) (4)(3)
(5)(1) (5)(2)
(21)(21) (6)(1)
(21)(31)
(31)(21)
Conjecture: Also the number of integer partitions y of n without a fixed point y(i) = i, ranked by A352826. This is stated at A238394, but Resta tells me he may not have had a proof. The a(2) = 1 through a(7) = 8 partitions without a fixed point are:
(2) (3) (4) (5) (6) (7)
(21) (31) (41) (33) (43)
(211) (311) (51) (61)
(2111) (411) (331)
(3111) (511)
(21111) (4111)
(31111)
(211111)
The version for compositions is A238351.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A238394 counts reversed partitions without a fixed point, ranked by A352830.
A238395 counts reversed partitions with a fixed point, ranked by A352872. (End)
The above conjecture is true. See Section 4 of the Blecher-Knopfmacher paper in the Links section. - Jeremy Lovejoy, Sep 26 2022
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REFERENCES
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B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 18 Entry 9 Corollary (i).
G. E. Andrews, B. C. Berndt, Ramanujan's Lost Notebook Part I, Springer, see p. 169 Entry 6.7.1.
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LINKS
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FORMULA
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G.f.: (Sum_{k>=0} (-1)^k * x^(k(k+1)/2)) / (Product_{k>0} 1-x^k). - Michael Somos, Jul 28 2003
G.f.: Sum_{i>=0} x^(i*(i+1)) / (Product_{j=1..i} 1-x^j )^2. - Jon Perry, Jul 18 2004
G.f.: (Sum_{i>=0} x^i / (Product_{j=1..i} 1-x^j)^2 ) * (Product_{k>0} 1-x^k). - Li Han, May 23 2020
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EXAMPLE
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G.f. = 1 + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 12*x^8 + 16*x^9 + 23*x^10 + ... - Michael Somos, Jan 15 2018
The a(0) = 1 through a(8) = 12 partitions with nonnegative crank:
() . (2) (3) (4) (5) (6) (7) (8)
(21) (22) (32) (33) (43) (44)
(31) (41) (42) (52) (53)
(221) (51) (61) (62)
(222) (322) (71)
(321) (331) (332)
(421) (422)
(2221) (431)
(521)
(2222)
(3221)
(3311)
(End)
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MATHEMATICA
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a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ (-1)^k x^(k (k + 1)/2) , {k, 0, (Sqrt[1 + 8 n] - 1)/2}] / QPochhammer[ x], {x, 0, n}]]; (* Michael Somos, Jan 15 2018 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(k (k + 1)) / QPochhammer[ x, x, k]^2 , {k, 0, (Sqrt[1 + 4 n] - 1)/2}], {x, 0, n}]]; (* Michael Somos, Jan 15 2018 *)
ck[y_]:=With[{w=Count[y, 1]}, If[w==0, If[y=={}, 0, Max@@y], Count[y, _?(#>w&)]-w]]; Table[Length[Select[IntegerPartitions[n], ck[#]>=0&]], {n, 0, 30}] (* Gus Wiseman, Mar 30 2021 *)
ici[q_]:=And@@Table[q[[i]]>q[[i+2]], {i, Length[q]-2}];
Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n], EvenQ@*Length], ici]], {n, 0, 15}] (* Gus Wiseman, Mar 30 2021 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(1 + 8*n) -1)\2, (-1)^k * x^((k+k^2)/2)) / eta( x + x * O(x^n)), n))}; /* Michael Somos, Jul 28 2003 */
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CROSSREFS
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These are the row-sums of the right (or left) half of A064391, inclusive.
These partitions are ranked by A352873.
A034008 counts even-length compositions.
A224958 counts compositions w/ alternating parts unequal (even: A342532).
A257989 gives the crank of the partition with Heinz number n.
A342527 counts compositions w/ alternating parts equal (even: A065608).
A342528 = compositions w/ alternating parts weakly decr. (even: A114921).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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