Displaying 1-10 of 12 results found.
Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with k parts p at position p (fixed points), n>=0, 0<=k<=n.
+10
43
1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 3, 4, 1, 0, 0, 6, 7, 3, 0, 0, 0, 11, 16, 4, 1, 0, 0, 0, 22, 29, 12, 1, 0, 0, 0, 0, 42, 60, 23, 3, 0, 0, 0, 0, 0, 82, 120, 47, 7, 0, 0, 0, 0, 0, 0, 161, 238, 100, 12, 1, 0, 0, 0, 0, 0, 0, 316, 479, 198, 30, 1, 0, 0, 0, 0, 0, 0, 0, 624, 956, 404, 61, 3, 0, 0, 0, 0, 0, 0, 0, 0, 1235, 1910, 818, 126, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0
COMMENTS
In general, column k is asymptotic to c(k) * 2^n. The constants c(k) numerically:
c(0) = 0.144394047543301210639449860964615390044455952420342... = A048651/2 c(1) = 0.231997216225445223894202367545783700531838988546098... = c(0)* A065442c(2) = 0.104261929557371534733906196116707679501974368826074...
c(3) = 0.017956317806894073430249112172514186063327165575720...
c(4) = 0.001343254222922697613125145839110293324517874530073...
c(5) = 0.000046459767012163920051487037952792359225887287888...
c(6) = 0.000000768651747857094917953943327540619110335556499...
c(7) = 0.000000006200599904985793344094393321042983316604040...
c(8) = 0.000000000024656652167851516076173236693314090168122...
c(9) = 0.000000000000048633746319332356416193899916110113745...
c(10)= 0.000000000000000047750743608910618576944191079881479...
c(20)= 1.05217230403079700467566...*10^(-63)
For big k is c(k) ~ m * 2^(-k*(k+1)/2), where m = 1/(4*c(0)) = 1/(2* A048651) = 1.7313733097275318... (End)
REFERENCES
M. Archibald, A. Blecher and A. Knopfmacher, Fixed points in compositions and words, accepted by the Journal of Integer Sequences.
EXAMPLE
Triangle starts:
00: 1,
01: 0, 1,
02: 1, 1, 0,
03: 2, 1, 1, 0,
04: 3, 4, 1, 0, 0,
05: 6, 7, 3, 0, 0, 0,
06: 11, 16, 4, 1, 0, 0, 0,
07: 22, 29, 12, 1, 0, 0, 0, 0,
08: 42, 60, 23, 3, 0, 0, 0, 0, 0,
09: 82, 120, 47, 7, 0, 0, 0, 0, 0, 0,
10: 161, 238, 100, 12, 1, 0, 0, 0, 0, 0, 0,
11: 316, 479, 198, 30, 1, 0, 0, 0, 0, 0, 0, 0,
12: 624, 956, 404, 61, 3, 0, 0, 0, 0, 0, 0, 0, 0,
13: 1235, 1910, 818, 126, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0,
14: 2449, 3817, 1652, 258, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
15: 4864, 7633, 3319, 537, 30, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
...
Row n = 5 counts the following compositions (empty columns indicated by dots):
(5) (14) (113) . . .
(23) (32) (122)
(41) (131) (1211)
(212) (221)
(311) (1112)
(2111) (1121)
(11111)
(End)
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, expand(
add(b(n-j, i+1)*`if`(i=j, x, 1), j=1..n)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 1)):
seq(T(n), n=0..15);
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == 0, 1, Expand[Sum[b[n-j, i+1]*If[i == j, x, 1], {j, 1, n}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 1]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Jan 06 2015, after Alois P. Heinz *) pq[y_]:=Length[Select[Range[Length[y]], #==y[[#]]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], pq[#]==k&]], {n, 0, 9}, {k, 0, n}] (* Gus Wiseman, Apr 03 2022 *)
CROSSREFS
Columns k=0-10 give: A238351, A240736, A240737, A240738, A240739, A240740, A240741, A240742, A240743, A240744, A240745. The version for permutations is A008290. The version with all zeros removed is A238350. The version for reversed partitions is A238352. Below: comps = compositions, first = column k=0, stat = rank statistic.
Number of integer compositions of n with exactly k nonfixed points (parts not on the diagonal).
+10
25
1, 1, 0, 0, 2, 0, 1, 1, 2, 0, 0, 4, 2, 2, 0, 0, 5, 5, 4, 2, 0, 1, 3, 12, 8, 6, 2, 0, 0, 7, 14, 19, 14, 8, 2, 0, 0, 8, 21, 33, 32, 22, 10, 2, 0, 0, 9, 30, 54, 63, 54, 32, 12, 2, 0, 1, 6, 47, 80, 116, 116, 86, 44, 14, 2, 0, 0, 11, 53, 129, 194, 229, 202, 130, 58, 16, 2, 0
COMMENTS
A nonfixed point in a composition c is an index i such that c_i != i.
EXAMPLE
Triangle begins:
1
1 0
0 2 0
1 1 2 0
0 4 2 2 0
0 5 5 4 2 0
1 3 12 8 6 2 0
0 7 14 19 14 8 2 0
0 8 21 33 32 22 10 2 0
0 9 30 54 63 54 32 12 2 0
1 6 47 80 116 116 86 44 14 2 0
For example, row n = 6 counts the following compositions (empty column indicated by dot):
(123) (6) (24) (231) (2112) (21111) .
(15) (33) (312) (2121) (111111)
(42) (51) (411) (3111)
(114) (1113) (11112)
(132) (1122) (11121)
(141) (1311) (11211)
(213) (2211)
(222) (12111)
(321)
(1131)
(1212)
(1221)
MATHEMATICA
pnq[y_]:=Length[Select[Range[Length[y]], #!=y[[#]]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], pnq[#]==k&]], {n, 0, 9}, {k, 0, n}]
CROSSREFS
The version for permutations is A098825. The corresponding rank statistic is A352513.
Number of compositions (p0, p1, p2, ...) of n with pi - p0 <= i and pi >= p0.
+10
24
1, 1, 2, 3, 5, 8, 14, 25, 46, 87, 167, 324, 634, 1248, 2466, 4887, 9706, 19308, 38455, 76659, 152925, 305232, 609488, 1217429, 2432399, 4860881, 9715511, 19421029, 38826059, 77626471, 155211785, 310357462, 620608652, 1241046343, 2481817484, 4963191718, 9925669171, 19850186856, 39698516655, 79394037319
COMMENTS
a(0)=1, otherwise row sums of A179748. For n>=1 cumulative sums of A008930. Also the number of integer compositions of n with exactly one part on or above the diagonal. For example, the a(1) = 1 through a(5) = 8 compositions are:
(1) (2) (3) (4) (5)
(11) (21) (31) (41)
(111) (112) (212)
(211) (311)
(1111) (1112)
(1121)
(2111)
(11111)
(End)
FORMULA
G.f.: 1 + q/(1-q) * sum(n>=0, q^n * prod(k=1..n, (1-q^k)/(1-q) ) ). [ Joerg Arndt, Mar 24 2014]
EXAMPLE
The a(7) = 25 such compositions are:
01: [ 1 1 1 1 1 1 1 ]
02: [ 1 1 1 1 1 2 ]
03: [ 1 1 1 1 2 1 ]
04: [ 1 1 1 1 3 ]
05: [ 1 1 1 2 1 1 ]
06: [ 1 1 1 2 2 ]
07: [ 1 1 1 3 1 ]
08: [ 1 1 1 4 ]
09: [ 1 1 2 1 1 1 ]
10: [ 1 1 2 1 2 ]
11: [ 1 1 2 2 1 ]
12: [ 1 1 2 3 ]
13: [ 1 1 3 1 1 ]
14: [ 1 1 3 2 ]
15: [ 1 2 1 1 1 1 ]
16: [ 1 2 1 1 2 ]
17: [ 1 2 1 2 1 ]
18: [ 1 2 1 3 ]
19: [ 1 2 2 1 1 ]
20: [ 1 2 2 2 ]
21: [ 1 2 3 1 ]
22: [ 2 2 3 ]
23: [ 2 3 2 ]
24: [ 3 4 ]
25: [ 7 ]
(End)
MAPLE
A179748 := proc(n, k) option remember; if k= 1 then 1; elif k> n then 0 ; else add( procname(n-i, k-1), i=1..k-1) ; end if; end proc:
MATHEMATICA
Clear[t, nn]; nn = 39; t[n_, 1] = 1; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, k - 1}], 0]; Table[Sum[t[n, k], {k, 1, n}], {n, 1, nn}] (* Mats Granvik, Jan 01 2015 *) pdw[y_]:=Length[Select[Range[Length[y]], #<=y[[#]]&]]; Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], pdw[#]==1&]], {n, 0, 10}] (* Gus Wiseman, Mar 31 2022 *)
PROG
(Sage)
@CachedFunction
if n == 0: return int(k==0);
if k == 1: return int(n>=1);
return sum( T(n-i, k-1) for i in [1..k-1] );
# to display triangle A179748 including column zero = [1, 0, 0, 0, ...]: #for n in [0..10]: print([ T(n, k) for k in [0..n] ])
def a(n): return sum( T(n, k) for k in [0..n] )
print([a(n) for n in [0..66]])
(PARI) N=66; q='q+O('q^N); Vec( 1 + q/(1-q) * sum(n=0, N, q^n * prod(k=1, n, (1-q^k)/(1-q) ) ) ) \\ Joerg Arndt, Mar 24 2014
CROSSREFS
Cf. A238859 (compositions with subdiagonal growth), A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth). Cf. A238860 (partitions with superdiagonal growth), A238861 (compositions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition). A352517 counts weak excedances of standard compositions.
Cf. A008930, A010054, A088218, A098825, A114088, A219282, A238352, A319005, A350839, A352488, A352489, A352514, A352515, A352516.
EXTENSIONS
New name and a(0) = 1 prepended, Joerg Arndt, Mar 24 2014
Number of fixed points in the n-th composition in standard order.
+10
20
0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 1, 0, 1, 2, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 2, 2, 2, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 3, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1
COMMENTS
The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. See also A000120, A059893, A070939, A114994, A225620. A fixed point of composition c is an index i such that c_i = i.
EXAMPLE
The 169th composition in standard order is (2,2,3,1), with fixed points {2,3}, so a(169) = 2.
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
pq[y_]:=Length[Select[Range[Length[y]], #==y[[#]]&]];
Table[pq[stc[n]], {n, 0, 100}]
Number of nonfixed points in the n-th composition in standard order.
+10
20
0, 0, 1, 1, 1, 2, 0, 2, 1, 2, 1, 3, 1, 1, 2, 3, 1, 2, 1, 3, 2, 2, 3, 4, 1, 2, 1, 2, 1, 3, 3, 4, 1, 2, 1, 3, 2, 2, 3, 4, 2, 3, 2, 3, 2, 4, 4, 5, 1, 2, 2, 3, 0, 2, 2, 3, 2, 2, 3, 4, 3, 4, 4, 5, 1, 2, 1, 3, 2, 2, 3, 4, 2, 3, 2, 3, 2, 4, 4, 5, 2, 3, 3, 4, 1, 3, 3
COMMENTS
The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. See also A000120, A059893, A070939, A114994, A225620. A nonfixed point in a composition c is an index i such that c_i != i.
EXAMPLE
The 169th composition in standard order is (2,2,3,1), with nonfixed points {1,4}, so a(169) = 2.
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
pnq[y_]:=Length[Select[Range[Length[y]], #!=y[[#]]&]];
Table[pnq[stc[n]], {n, 0, 100}]
Triangle read by rows where T(n,k) is the number of integer compositions of n with k weak nonexcedances (parts on or below the diagonal).
+10
20
1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 2, 3, 4, 3, 3, 1, 3, 4, 8, 6, 6, 4, 1, 4, 7, 12, 13, 12, 10, 5, 1, 5, 13, 16, 26, 24, 22, 15, 6, 1, 7, 19, 27, 43, 48, 46, 37, 21, 7, 1, 10, 26, 47, 68, 90, 93, 83, 58, 28, 8, 1, 14, 36, 77, 109, 159, 180, 176, 141
EXAMPLE
Triangle begins:
1
0 1
1 0 1
1 1 1 1
1 3 1 2 1
2 3 4 3 3 1
3 4 8 6 6 4 1
4 7 12 13 12 10 5 1
5 13 16 26 24 22 15 6 1
7 19 27 43 48 46 37 21 7 1
10 26 47 68 90 93 83 58 28 8 1
For example, row n = 6 counts the following compositions:
(6) (15) (114) (123) (1113) (11112) (111111)
(24) (42) (132) (1311) (1122) (11121)
(33) (51) (141) (2112) (1131) (11211)
(231) (213) (2121) (1212) (12111)
(222) (2211) (1221)
(312) (3111) (21111)
(321)
(411)
MATHEMATICA
pw[y_]:=Length[Select[Range[Length[y]], #>=y[[#]]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], pw[#]==k&]], {n, 0, 15}, {k, 0, n}]
PROG
(PARI) T(n)={my(v=vector(n+1, i, i==1), r=v); for(k=1, n, v=vector(#v, j, sum(i=1, j-1, if(k>=i, x, 1)*v[j-i])); r+=v); [Vecrev(p) | p<-r]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 19 2023
CROSSREFS
The strong version for partitions is A114088. The opposite version for partitions is A115994. The corresponding rank statistic is A352515. A008292 is the triangle of Eulerian numbers (version without zeros).
Irregular triangle read by rows where T(n,k) is the number of integer compositions of n with k weak excedances (parts on or above the diagonal), all zeros removed.
+10
20
1, 1, 2, 3, 1, 5, 3, 8, 8, 14, 17, 1, 25, 35, 4, 46, 70, 12, 87, 137, 32, 167, 268, 76, 1, 324, 525, 170, 5, 634, 1030, 367, 17, 1248, 2026, 773, 49, 2466, 3999, 1598, 129, 4887, 7914, 3267, 315, 1, 9706, 15695, 6631, 730, 6, 19308, 31181, 13393, 1631, 23
EXAMPLE
Triangle begins:
1
1
2
3 1
5 3
8 8
14 17 1
25 35 4
46 70 12
87 137 32
167 268 76 1
324 525 170 5
For example, row n = 6 counts the following compositions:
(6) (15) (123)
(51) (24)
(312) (33)
(411) (42)
(1113) (114)
(1122) (132)
(2112) (141)
(2121) (213)
(3111) (222)
(11112) (231)
(11121) (321)
(11211) (1131)
(21111) (1212)
(111111) (1221)
(1311)
(2211)
(12111)
MATHEMATICA
pdw[y_]:=Length[Select[Range[Length[y]], #<=y[[#]]&]];
DeleteCases[Table[Length[Select[Join@@ Permutations/@IntegerPartitions[n], pdw[#]==k&]], {n, 0, 10}, {k, 0, n}], 0, {2}]
PROG
(PARI) T(n)={my(v=vector(n+1, i, i==1), r=v); for(k=1, n, v=vector(#v, j, sum(i=1, j-1, if(k<=i, x, 1)*v[j-i])); r+=v); r[1]=x; [Vecrev(p) | p<-r/x]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 19 2023
CROSSREFS
The version for partitions is A115994. The corresponding rank statistic is A352517. A008292 is the triangle of Eulerian numbers (version without zeros).
Triangle read by rows where T(n,k) is the number of integer compositions of n with k strong nonexcedances (parts below the diagonal).
+10
18
1, 1, 0, 1, 1, 0, 2, 1, 1, 0, 3, 2, 2, 1, 0, 4, 5, 3, 3, 1, 0, 6, 8, 7, 6, 4, 1, 0, 9, 12, 15, 12, 10, 5, 1, 0, 13, 19, 27, 25, 22, 15, 6, 1, 0, 18, 32, 43, 51, 46, 37, 21, 7, 1, 0, 25, 51, 70, 94, 94, 83, 58, 28, 8, 1, 0, 35, 77, 117, 162, 184, 176, 141, 86, 36, 9, 1, 0
EXAMPLE
Triangle begins:
1
1 0
1 1 0
2 1 1 0
3 2 2 1 0
4 5 3 3 1 0
6 8 7 6 4 1 0
9 12 15 12 10 5 1 0
13 19 27 25 22 15 6 1 0
18 32 43 51 46 37 21 7 1 0
25 51 70 94 94 83 58 28 8 1 0
For example, row n = 6 counts the following compositions (empty column indicated by dot):
(6) (51) (312) (1113) (11112) (111111) .
(15) (114) (411) (1122) (11121)
(24) (132) (1131) (2112) (11211)
(33) (141) (1212) (2121) (21111)
(42) (213) (1221) (3111)
(123) (222) (1311) (12111)
(231) (2211)
(321)
MATHEMATICA
pa[y_]:=Length[Select[Range[Length[y]], #>y[[#]]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], pa[#]==k&]], {n, 0, 15}, {k, 0, n}]
PROG
(PARI) T(n)={my(v=vector(n+1, i, i==1), r=v); for(k=1, n, v=vector(#v, j, sum(i=1, j-1, if(k>i, x, 1)*v[j-i])); r+=v); vector(#v, i, Vecrev(r[i], i))}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 19 2023
CROSSREFS
The version for partitions is A114088. Row sums without the last term are A131577. The version for permutations is A173018. The corresponding rank statistic is A352514. A008292 is the triangle of Eulerian numbers (version without zeros).
Irregular triangle read by rows where T(n,k) is the number of integer compositions of n with k excedances (parts above the diagonal), all zeros removed.
+10
18
1, 1, 1, 1, 2, 2, 3, 5, 6, 9, 1, 11, 18, 3, 21, 35, 8, 41, 67, 20, 80, 131, 44, 1, 157, 257, 94, 4, 310, 505, 197, 12, 614, 996, 406, 32, 1218, 1973, 825, 80, 2421, 3915, 1669, 186, 1, 4819, 7781, 3364, 415, 5, 9602, 15486, 6762, 901, 17, 19147, 30855, 13567, 1918, 49
EXAMPLE
Triangle begins:
1
1
1 1
2 2
3 5
6 9 1
11 18 3
21 35 8
41 67 20
80 131 44 1
157 257 94 4
310 505 197 12
614 996 406 32
For example, row n = 5 counts the following compositions:
(113) (5) (23)
(122) (14)
(1112) (32)
(1121) (41)
(1211) (131)
(11111) (212)
(221)
(311)
(2111)
MATHEMATICA
pd[y_]:=Length[Select[Range[Length[y]], #<y[[#]]&]];
DeleteCases[Table[Length[Select[Join@@ Permutations/@IntegerPartitions[n], pd[#]==k&]], {n, 0, 10}, {k, 0, n}], 0, {2}]
PROG
(PARI)
S(v, u)={vector(#v, k, sum(i=1, k-1, v[k-i]*u[i]))}
T(n)={my(v=vector(1+n), s); v[1]=1; s=v; for(i=1, n, v=S(v, vector(n, j, if(j>i, 'x, 1))); s+=v); [Vecrev(p) | p<-s]}
{ my(A=T(12)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 02 2023
CROSSREFS
The opposite version for partitions is A114088. The weak version for partitions is A115994. The corresponding rank statistic is A352516.
Numbers whose weakly increasing prime indices y have exactly one fixed point y(i) = i.
+10
13
2, 4, 8, 9, 10, 12, 14, 16, 22, 24, 26, 27, 28, 32, 34, 36, 38, 40, 44, 46, 48, 52, 58, 60, 62, 63, 64, 68, 70, 72, 74, 75, 76, 80, 81, 82, 86, 88, 92, 94, 96, 98, 99, 104, 106, 108, 110, 112, 116, 117, 118, 120, 122, 124, 125, 128, 130, 132, 134, 135, 136
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
EXAMPLE
The terms together with their prime indices begin:
2: {1} 36: {1,1,2,2} 74: {1,12}
4: {1,1} 38: {1,8} 75: {2,3,3}
8: {1,1,1} 40: {1,1,1,3} 76: {1,1,8}
9: {2,2} 44: {1,1,5} 80: {1,1,1,1,3}
10: {1,3} 46: {1,9} 81: {2,2,2,2}
12: {1,1,2} 48: {1,1,1,1,2} 82: {1,13}
14: {1,4} 52: {1,1,6} 86: {1,14}
16: {1,1,1,1} 58: {1,10} 88: {1,1,1,5}
22: {1,5} 60: {1,1,2,3} 92: {1,1,9}
24: {1,1,1,2} 62: {1,11} 94: {1,15}
26: {1,6} 63: {2,2,4} 96: {1,1,1,1,1,2}
27: {2,2,2} 64: {1,1,1,1,1,1} 98: {1,4,4}
28: {1,1,4} 68: {1,1,7} 99: {2,2,5}
32: {1,1,1,1,1} 70: {1,3,4} 104: {1,1,1,6}
34: {1,7} 72: {1,1,1,2,2} 106: {1,16}
For example, 63 is in the sequence because its prime indices {2,2,4} have a unique fixed point at the second position.
MATHEMATICA
pq[y_]:=Length[Select[Range[Length[y]], #==y[[#]]&]];
Select[Range[100], pq[Flatten[Cases[FactorInteger[#], {p_, k_}:>Table[PrimePi[p], {k}]]]]==1&]
CROSSREFS
* = unproved
These are the positions of 1's in A352822. *The reverse version for no fixed points is A352826, counted by A064428. A238352 counts reversed partitions by fixed points.
A352833 counts partitions by fixed points.
Cf. A062457, A064410, A065770, A177510, A257990, A342192, A349158, A351983, A352520, A352823, A352824.
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