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A257989 The crank of the partition having Heinz number n.

%I A257989 #27 Apr 05 2021 21:07:39
%S A257989 -1,2,-2,3,0,4,-3,2,0,5,-2,6,0,3,-4,7,1,8,-1,4,0,9,-3,3,0,2,-1,10,1,
%T A257989 11,-5,5,0,4,-2,12,0,6,-3,13,1,14,-1,3,0,15,-4,4,1,7,-1,16,2,5,-2,8,0,
%U A257989 17,-1,18,0,4,-6,6,1,19,-1,9,1,20,-3,21,0,3,-1,5,1,22,-4,2,0,23,-1,7,0,10,-2,24,2,6,-1
%N A257989 The crank of the partition having Heinz number n.
%C A257989 The crank of a partition p is defined to be (i) the largest part of p if there is no 1 in p and (ii) (the number of parts larger than the number of 1's) minus (the number of 1's).
%C A257989 We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by _Alois P. Heinz_ in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436.
%C A257989 In the Maple program the subprogram B yields the partition with Heinz number n, the subprogram b yields the number of 1's in the partition with Heinz number n and the subprogram c yields the number of parts that are larger than the number of 1's in the partition with the Heinz number n.
%H A257989 Alois P. Heinz, <a href="/A257989/b257989.txt">Table of n, a(n) for n = 2..10000</a>
%H A257989 G. E. Andrews and F. Garvan, <a href="http://dx.doi.org/10.1090/S0273-0979-1988-15637-6">Dyson's crank of a partition</a>, Bull. Amer. Math. Soc., 18 (1988), 167-171.
%H A257989 B. C. Berndt, H. H. Chan, S. H. Chan, W.-C. Liaw, <a href="http://dx.doi.org/10.1016/j.jcta.2004.06.013">Cranks and dissections in Ramanujan's lost notebook</a>, J. Comb. Theory, Ser. A, 109, 2005, 91-120.
%H A257989 B. C. Berndt, H. H. Chan, S. H. Chan, W.-C. Liaw, <a href="http://www.math.uiuc.edu/~berndt/articles/finalproblem.pdf">Cranks - really the final problem</a>, Ramanujan J., 23, 2010, 3-15.
%H A257989 G. E. Andrews, K. Ono, <a href="http://dx.doi.org/10.1073/pnas.0507844102">Ramanujan's congruences and Dyson's crank</a>, Proc. Natl. Acad. Sci. USA, 102, 2005, 15277.
%H A257989 FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000474/">Dyson's crank of a partition.</a>
%H A257989 K. Mahlburg, <a href="http://dx.doi.org/10.1073/pnas.0506702102">Partition congruences and the Andrews-Garvan-Dyson crank</a>, Proc. Natl. Acad. Sci. USA, 102, 2005, 15373-15376.
%H A257989 Wikipedia, <a href="http://en.wikipedia.org/wiki/Crank_of_a_partition">Crank of a partition</a>
%e A257989 a(12) = - 2 because the partition with Heinz number 12 = 2*2*3 is [1,1,2], the number of parts larger than the number of 1's is 0 and the number of 1's is 2; 0 - 2 = -2.
%e A257989 a(945) = 4 because the partition with Heinz number 945 = 3^3 * 5 * 7 is [2,2,2,3,4] which has no part 1; the largest part is 4.
%e A257989 From _Gus Wiseman_, Apr 05 2021: (Start)
%e A257989 The partitions (center) with each Heinz number (left), and the corresponding terms (right):
%e A257989    2:    (1)    -> -1
%e A257989    3:    (2)    ->  2
%e A257989    4:   (1,1)   -> -2
%e A257989    5:    (3)    ->  3
%e A257989    6:   (2,1)   ->  0
%e A257989    7:    (4)    ->  4
%e A257989    8:  (1,1,1)  -> -3
%e A257989    9:   (2,2)   ->  2
%e A257989   10:   (3,1)   ->  0
%e A257989   11:    (5)    ->  5
%e A257989   12:  (2,1,1)  -> -2
%e A257989   13:    (6)    ->  6
%e A257989   14:   (4,1)   ->  0
%e A257989   15:   (3,2)   ->  3
%e A257989   16: (1,1,1,1) -> -4
%e A257989 (End)
%p A257989 with(numtheory): a := proc (n) local B, b, c: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do; [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: b := proc (n) if `mod`(n, 2) = 1 then 0 else 1+b((1/2)*n) end if end proc: c := proc (n) local b, B, ct, i: b := proc (n) if `mod`(n, 2) = 1 then 0 else 1+b((1/2)*n) end if end proc: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do: [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: ct := 0: for i to bigomega(n) do if b(n) < B(n)[i] then ct := ct+1 else  end if end do: ct end proc: if b(n) = 0 then max(B(n)) else c(n)-b(n) end if end proc: seq(a(n), n = 2 .. 150);
%t A257989 B[n_] := Module[{nn, j, m}, nn =  FactorInteger[n]; For[j = 1, j <= Length[nn], j++, m[j] = nn[[j]]]; Flatten[Table[Table[PrimePi[m[i][[1]]], {q, 1, m[i][[2]]}], {i, 1, Length[nn]}]]];
%t A257989 b[n_] := b[n] = If[OddQ[n], 0, 1 + b[n/2]];
%t A257989 c[n_] := Module[{ct, i}, ct = 0; For[i = 1, i <= PrimeOmega[n], i++, If[ b[n] < B[n][[i]], ct++]]; ct];
%t A257989 a[n_] := If[b[n] == 0, Max[B[n]], c[n] - b[n]];
%t A257989 Table[a[n], {n, 2, 100}] (* _Jean-François Alcover_, Apr 25 2017, after _Emeric Deutsch_ *)
%t A257989 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A257989 ck[y_]:=With[{w=Count[y,1]},If[w==0,Max@@y,Count[y,_?(#>w&)]-w]];
%t A257989 Table[ck[primeMS[n]],{n,2,30}] (* _Gus Wiseman_, Apr 05 2021 *)
%Y A257989 Cf. A215366, A257988.
%Y A257989 Indices of zeros are A342192.
%Y A257989 A001522 counts partitions of crank 0.
%Y A257989 A003242 counts anti-run compositions.
%Y A257989 A064391 counts partitions by crank.
%Y A257989 A064428 counts partitions of nonnegative crank.
%Y A257989 Cf. A000005, A000726, A056239, A112798, A124010, A224958, A325351, A325352.
%K A257989 sign
%O A257989 2,2
%A A257989 _Emeric Deutsch_, May 18 2015