C Count lacings of a shoe with two rows of N eyelets C subjected to the "physical" condition that all C eyelets are "loaded", i.e. that there is a force C component towards the opposite eyelet column. C C Author: Hugo Pfoertner http:/www.pfoertner.org/ C C Version history: C 05.01.2003 NEXPER replaced by LPG C 03.01.2003 Count crossing-free lacings (OEIS sequence A079410) C 17.12.2002 Initial version C C N is the number of eyelets in one row. C N=8 takes approx 1day CPU time on an Intel PIII 550MHz, C ((2*8-2)!=14! configs to check) C N=9 is expected to run approx 15*16=240days on the same CPU. C PARAMETER (N=4, N2=N+N, M=N2-2, N2P=N2+1, N2M=N2-1) PARAMETER (NN=N*N+N*N) PARAMETER ( EPS0=0.001D0, EPS1=0.999D0) INTEGER E(N2), EE(N,2), A(M), CC(0:NN) INTEGER LOVER DOUBLEPRECISION Q EQUIVALENCE (E,EE) C C FIXED START AND END EE(1,1) = 1 EE(N,2) = N2 C PRESET PERMUTATION ARRAY DO 2 I = 1, M A(I) = I 2 CONTINUE C STATUS OF PERMUTATION GENERATION NEXT = -1 C CONTER FOR LEGAL LACINGS L = 0 C OVERFLOW COUNTER IF L EXCEEDS 1E9 C THIS OCCURS FOR N>=8 C TOTAL COUNT: LOVER*1E9 + L LOVER = 0 C COUNTER FOR CROSSINGS DO 5 I = 0, NN CC(I) = 0 5 CONTINUE C WRITE (*,1002) N 1002 FORMAT ( ' N =', I2, // , 'Crossing-free Lacings:', / ) C 10 CONTINUE C CREATE NEXT PERMUTATION IF ( NEXT .EQ. -1 ) THEN NEXT = 1 ELSE C Create permutations in lexicographic order CALL LPG ( M, A, NEXT ) ENDIF IF ( NEXT .EQ. 0 ) GOTO 200 C INSERT PERMUTED PATH BETWEEN FIXED START AND END DO 20 I = 1, M E(I+1) = A(I) + 1 20 CONTINUE C TEST CONDITION THAT AT LEAST ONE OF THE DIRECTLY CONNECTED C EYELETS IS ON THE OPPOSITE SIDE DO 30 I = 2, N2M IF ( E(I) .LE. N ) THEN IF ( E(I-1) .GT. N ) GOTO 30 IF ( E(I+1) .GT. N ) GOTO 30 ELSE IF ( E(I-1) .LE. N ) GOTO 30 IF ( E(I+1) .LE. N ) GOTO 30 ENDIF C UNLOADED EYELET FOUND, ABANDON PATH GOTO 10 30 CONTINUE C C LEGAL LACING FOUND, CHECK FOR CROSSINGS C Left eyelet row at x=0, right row at x=1, C determine if intersection of lace path intervals C occurs between x=eps0 and x=eps1 C (eps0=0.001, eps1=0.999) C g1 given by two points (0,II),(1,JJ) C g2 given by two points (0,KK),(1,LL) C C K counts crossings K = 0 C DO 40 I = 2, N2M C IF BOTH ON SAME SIDE NO FURTHER CHECK NECESSARY IF ( E(I-1) .LE. N ) THEN C BOTH ON SAME SIDE, NO FURTHER CHECK IF ( E(I) .LE. N ) GOTO 40 II = E(I-1) JJ = N2P - E(I) ELSE C BOTH ON SAME SIDE, NO FURTHER CHECK IF ( E(I) .GT. N ) GOTO 40 II = E(I) JJ = N2P - E(I-1) ENDIF DO 50 J = I, N2M IF ( E(J) .LE. N ) THEN C BOTH ON SAME SIDE, NO FURTHER CHECK IF ( E(J+1) .LE. N ) GOTO 50 KK = E(J) LL = N2P - E(J+1) ELSE C BOTH ON SAME SIDE, NO FURTHER CHECK IF ( E(J+1) .GT. N ) GOTO 50 KK = E(J+1) LL = N2P - E(J) ENDIF C DIAGNOSTIC OUTPUT C WRITE (*,*) II, JJ, KK, LL NUM = II - KK NEN = LL - JJ + NUM IF ( NEN .EQ. 0 ) GOTO 50 Q = DBLE(NUM)/DBLE(NEN) IF ( Q .LE. EPS0 ) GOTO 50 IF ( Q .GE. EPS1 ) GOTO 50 C INCREASE CROSS COUNT K = K + 1 50 CONTINUE 40 CONTINUE C COUNT LACINGS L = L + 1 IF ( L .GT. 1000000000 ) THEN L = L - 1000000000 LOVER = LOVER + 1 WRITE (*,*) ' L OVERFLOW: ', LOVER ENDIF C TO PRINT RESULTS ACTIVATE COMMENTED LINES C PRINT CROSSING-FREE CONFIGS (deactivate for N=8) IF ( K .EQ. 0 ) & WRITE (*,1000) L, K, (E(I),I=1,N2) 1000 FORMAT (I8, I2, 2I3,:,2I3,:,2I3,:,2I3,:,2I3,:,2I3,:,2I3,:,2I3) C UPDATE CROSSING COUNT HISTOGRAM CC(K) = CC(K) + 1 IF ( NEXT .EQ. 1 ) GOTO 10 C 200 CONTINUE WRITE (*,*) ' LOVER=', LOVER, ' L=', L WRITE (*,*) ' Crossings, #' DO 100 I = 0, NN IF ( CC(I) .GT. 0 ) WRITE (*,1001) I, CC(I) 1001 FORMAT ( I3, I10 ) 100 CONTINUE END C******************************************************************** SUBROUTINE LPG ( N, A, NEXT ) C C Generate next permutation of N elements A(i) in C lexicographic order C C The method used is C "Algorithm L (Lexicographic permutation generation)" C described in Chapter 7.2.1.2 C with the modification proposed in the answer to C exercise 1 [J.P.N. Phillips, Comp. J. 10 (1967), 311)] C of Donald E. Knuth's The Art of Computer Programming, Volume 4, C Combinatorial Algorithms, C Volume 4A, Enumeration and Backtracking. C The printed book is expected to be available in the year 2007. C D. Knuth has put some chapters online, including C Pre-fascicle 2b (generating all permutations), available at C http://www-cs-faculty.stanford.edu/~knuth/fasc2b.ps.gz C Most of the comments are copied verbally from Knuth's C description C C Author: Hugo Pfoertner http://www.pfoertner.org/ C C Change history: C 06.01.2003 Optional code to treat N<4 C 05.01.2003 Initial version C C Call parameters: INTEGER N, A(N), NEXT C N: number of elements to permute. N>=4 must hold (not checked) C A: Integer array holding the elements. C to be preset externally (e.g. by 1,2,3,...,n-1,n) C A is overwritten by the lexicographically next permutation C NEXT: =1 if A has been successfully updated C =0 if A was already in reverse lexicographic order C i.e. if a(n)>=a(n-1)>=....>=a(2)>=a(1) C In this case no change is made to A C C Local variables INTEGER J, K, L, X, Y, Z, H C C Preset return status NEXT = 1 C C In a generally applicable version for all N all lines starting C with comment CS have to be activated CSC Start part necessary for N<3 CS IF ( N .LE. 1 ) THEN CS NEXT = 0 CS RETURN CS ENDIF CS IF ( N .EQ. 2 ) THEN CS IF ( A(1) .LT. A(2) ) THEN CS H = A(1) CS A(1) = A(2) CS A(2) = H CS ELSE CS NEXT = 0 CS ENDIF CS RETURN CS ENDIF CSC End part necessary for N<3 C C Find the largest j such that a(j) can be increased C C Step L2 finds j=n-1 half of the time when the elements are distinct, C because exactly n!/2 of the n! permutations have a(n-1)= a(l) decrease l repeatedly until a(j)a(l) C C L3' Easy increase? IF ( Y .LT. Z ) THEN A(J) = Z A(J+1) = Y A(N) = X C Done, go to L4.1' GOTO 41 ENDIF C L3.1' Increase A(J) L = N - 1 20 CONTINUE IF ( Y .GE. A(L) ) THEN L = L - 1 GOTO 20 ENDIF C C Since a(j+1) >= ... >= a(n), element a(l) is the smallest element C greater than a(j) that can legitimately follow a(1)...a(j-1) in a C permutation. Before the interchange we have a(j+1)>=...>=a(l-1)>=a(l) C >a(j)>=a(l+1)>=...>=a(n) C C Interchange C A(J) = A(L) A(L) = Y C C After the interchange, we have C a(j+1)>=...>=a(l-1)>=a(j)>a(l)>=a(l+1)>=...>=a(n) C C Find the lexicographically least way to extend the new a(1)...a(j) C to a complete pattern: C The first permutation beginning with the current prefix a(1)...a(j) C is a(1)...a(j)a(n)...a(j+1), and step L4 produces it by doing C floor (n-j)/2 interchanges C C L4 Reverse a(j+1)...a(n) C L4' Begin to reverse A(N) = A(J+1) A(J+1) = Z C L4.1' Reverse a(j+1)...a(n-1) 41 CONTINUE K = J + 2 L = N - 1 30 CONTINUE IF ( K .LT. L ) THEN H = A(K) A(K) = A(L) A(L) = H K = K + 1 L = L - 1 GOTO 30 ENDIF RETURN C End of SUBROUTINE LPG END --------------------------------------------------------------------- Results, edited + some ASCII illustrations N = 3 Crossing-free Lacings: 3 0 1 2 5 3 4 6 5 0 1 3 4 2 5 6 6 0 1 3 4 5 2 6 15 0 1 5 2 3 4 6 L= 20 (=A078698(n)) 3 5 6 15 1 6 1 6 1 6 1 6 | | | | | / \ | | | | | | / \ | | | | | | / \ | 2___5| |2___5 |2___5 2___5| / | | \ | | | | / | | \ | | | | / | | \ | | | | 3____4 3____4 3____4 3____4 Crossings, # 0 4 =2*A079410(3) 1 7 2 5 3 3 6 1 (6=A000384(2)) --------------------------------------------------------------------- N = 4 Crossing-free Lacings: 35 0 1 2 7 3 6 4 5 8 43 0 1 2 7 5 4 3 6 8 45 0 1 2 7 6 3 4 5 8 70 0 1 3 6 4 5 7 2 8 75 0 1 3 6 5 4 2 7 8 77 0 1 3 6 7 2 4 5 8 101 0 1 4 5 3 6 2 7 8 102 0 1 4 5 3 6 7 2 8 107 0 1 4 5 6 3 2 7 8 108 0 1 4 5 6 3 7 2 8 109 0 1 4 5 7 2 3 6 8 315 0 1 7 2 3 6 4 5 8 317 0 1 7 2 4 5 3 6 8 327 0 1 7 2 6 3 4 5 8 L= 396 35 43 45 70 77 102 108 1 8 1 8 1 8 1 8 1 8 1 8 1 8 | | | | | | | / | | | / | / | | | | | | | / | | | / | / | | | | | | | / | | | / | / 2___7| 2___7| 2___7| |2___7 |2__7| |2___7 |2___7 / | \/ || | | \/ || | | | / / | /\ || | | /\ || | | | / / | || || | | || || | | | / 3___6| 3___6| 3___6| 3___6| |3__6| |3___6 |3___6 / | | | | | / | | | | \ | | / | | | | | / | | | | \ | | / | | | | | / | | | | \ | | 4____5 4____5 4____5 4____5 4____5 4____5 4____5 109 75 107 317 109 315 327 1 8 1 8 1 8 1 8 1 8 1 8 1 8 | | | | | | \ | | | \ | \ | | | | | | | \ | | | \ | \ | | | | | | | \ | | | \ | \ | |2___7 |2___7 |2___7 2___7| |2__7| 2___7| 2___7| | \ \/ || | | || \/ | | \ | | \ /\ || | | || /\ | | \ | | \ || || | | || || | | \ | |3___6 |3___6 |3___6 |3___6 |3__6| 3___6| 3___6| | \ | | | | | \ | | / | | | | \ | | | | | \ | | / | | | | \ | | | | | \ | | / | | | 4____5 4____5 4____5 4____5 4____5 4____5 4____5 Crossings, # 0 14 =2*A079410(4) 1 36 2 51 3 69 4 84 5 52 6 34 7 29 8 13 9 6 11 3 12 4 15 1 (15=A000384(3)) -------------------------------------------------------------------- N = 5 Crossing-free Lacings: 1085 0 1 2 9 3 4 8 6 5 7 10 1086 0 1 2 9 3 4 8 7 5 6 10 1087 0 1 2 9 3 5 6 7 4 8 10 1088 0 1 2 9 3 5 6 8 4 7 10 1127 0 1 2 9 3 8 4 7 5 6 10 1135 0 1 2 9 3 8 6 5 4 7 10 1137 0 1 2 9 3 8 7 4 5 6 10 1303 0 1 2 9 6 5 3 8 4 7 10 1305 0 1 2 9 6 5 4 7 3 8 10 1311 0 1 2 9 6 5 7 4 3 8 10 1351 0 1 2 9 7 4 6 5 3 8 10 1355 0 1 2 9 7 4 8 3 5 6 10 1383 0 1 2 9 8 3 4 7 5 6 10 1385 0 1 2 9 8 3 5 6 4 7 10 1395 0 1 2 9 8 3 7 4 5 6 10 1867 0 1 3 7 4 5 6 8 2 9 10 1868 0 1 3 7 4 5 6 9 2 8 10 2215 0 1 3 8 2 9 6 5 4 7 10 2217 0 1 3 8 2 9 7 4 5 6 10 2225 0 1 3 8 4 2 9 6 5 7 10 2226 0 1 3 8 4 2 9 7 5 6 10 2254 0 1 3 8 4 7 5 6 9 2 10 2259 0 1 3 8 4 7 6 5 2 9 10 2261 0 1 3 8 4 7 9 2 5 6 10 2386 0 1 3 8 6 5 4 7 9 2 10 2391 0 1 3 8 6 5 7 4 2 9 10 2426 0 1 3 8 7 4 5 6 9 2 10 2431 0 1 3 8 7 4 6 5 2 9 10 2463 0 1 3 8 9 2 4 7 5 6 10 2465 0 1 3 8 9 2 5 6 4 7 10 2525 0 1 3 9 2 4 8 6 5 7 10 2526 0 1 3 9 2 4 8 7 5 6 10 2527 0 1 3 9 2 5 6 7 4 8 10 2528 0 1 3 9 2 5 6 8 4 7 10 2754 0 1 3 9 6 5 7 8 4 2 10 2800 0 1 3 9 7 4 8 6 5 2 10 2814 0 1 3 9 7 5 6 8 4 2 10 2860 0 1 3 9 8 4 7 6 5 2 10 3005 0 1 4 6 5 2 9 7 3 8 10 3006 0 1 4 6 5 2 9 8 3 7 10 3007 0 1 4 6 5 3 7 8 2 9 10 3008 0 1 4 6 5 3 7 9 2 8 10 3307 0 1 4 7 3 5 6 8 2 9 10 3308 0 1 4 7 3 5 6 9 2 8 10 3339 0 1 4 7 3 8 6 5 2 9 10 3341 0 1 4 7 3 8 9 2 5 6 10 3383 0 1 4 7 5 6 8 3 2 9 10 3384 0 1 4 7 5 6 8 3 9 2 10 3385 0 1 4 7 5 6 9 2 3 8 10 3463 0 1 4 7 6 5 2 9 3 8 10 3465 0 1 4 7 6 5 3 8 2 9 10 3466 0 1 4 7 6 5 3 8 9 2 10 3506 0 1 4 7 8 3 5 6 9 2 10 3515 0 1 4 7 8 3 9 2 5 6 10 3520 0 1 4 7 8 3 9 6 5 2 10 3555 0 1 4 7 9 2 8 3 5 6 10 3580 0 1 4 7 9 3 8 6 5 2 10 3665 0 1 4 8 3 2 9 6 5 7 10 3666 0 1 4 8 3 2 9 7 5 6 10 3834 0 1 4 8 6 5 7 9 3 2 10 3894 0 1 4 8 7 5 6 9 3 2 10 4445 0 1 5 6 4 2 9 7 3 8 10 4446 0 1 5 6 4 2 9 8 3 7 10 4447 0 1 5 6 4 3 7 8 2 9 10 4448 0 1 5 6 4 3 7 9 2 8 10 4457 0 1 5 6 4 7 3 8 2 9 10 4458 0 1 5 6 4 7 3 8 9 2 10 4463 0 1 5 6 4 7 8 3 2 9 10 4464 0 1 5 6 4 7 8 3 9 2 10 4465 0 1 5 6 4 7 9 2 3 8 10 4543 0 1 5 6 7 4 2 9 3 8 10 4545 0 1 5 6 7 4 3 8 2 9 10 4546 0 1 5 6 7 4 3 8 9 2 10 4551 0 1 5 6 7 4 8 3 2 9 10 4552 0 1 5 6 7 4 8 3 9 2 10 4554 0 1 5 6 7 4 8 9 3 2 10 4586 0 1 5 6 8 3 4 7 9 2 10 4591 0 1 5 6 8 3 7 4 2 9 10 4595 0 1 5 6 8 3 9 2 4 7 10 4600 0 1 5 6 8 3 9 7 4 2 10 4614 0 1 5 6 8 4 7 9 3 2 10 4623 0 1 5 6 9 2 3 8 4 7 10 4625 0 1 5 6 9 2 4 7 3 8 10 4635 0 1 5 6 9 2 8 3 4 7 10 4660 0 1 5 6 9 3 8 7 4 2 10 12707 0 1 9 2 3 8 4 7 5 6 10 12715 0 1 9 2 3 8 6 5 4 7 10 12717 0 1 9 2 3 8 7 4 5 6 10 12747 0 1 9 2 4 7 6 5 3 8 10 12749 0 1 9 2 4 7 8 3 5 6 10 12773 0 1 9 2 5 6 4 7 3 8 10 12779 0 1 9 2 5 6 7 4 3 8 10 12781 0 1 9 2 5 6 8 3 4 7 10 12987 0 1 9 2 8 3 4 7 5 6 10 12989 0 1 9 2 8 3 5 6 4 7 10 12999 0 1 9 2 8 3 7 4 5 6 10 14443 0 1 9 6 5 2 8 3 4 7 10 14451 0 1 9 6 5 3 7 4 2 8 10 14455 0 1 9 6 5 3 8 2 4 7 10 14463 0 1 9 6 5 4 7 3 2 8 10 14571 0 1 9 7 3 4 6 5 2 8 10 14583 0 1 9 7 3 5 6 4 2 8 10 14623 0 1 9 7 4 2 8 3 5 6 10 14635 0 1 9 7 4 3 8 2 5 6 10 14743 0 1 9 8 2 3 7 4 5 6 10 14751 0 1 9 8 2 4 6 5 3 7 10 14755 0 1 9 8 2 4 7 3 5 6 10 14763 0 1 9 8 2 5 6 4 3 7 10 L= 14976 Crossings, # 0 108 =2*A079410(5) 1 328 2 650 3 1035 4 1441 5 1602 6 1640 7 1505 8 1437 9 1332 10 1027 11 904 12 634 13 535 14 327 15 166 16 94 17 67 18 55 19 14 20 20 21 26 22 17 24 5 25 6 28 1 (28=A000384(4)) -------------------------------------------------------------------- N = 6 Crossing-free Lacings: 55409 0 1 2 11 3 4 10 5 9 7 6 8 12 55410 0 1 2 11 3 4 10 5 9 8 6 7 12 55421 0 1 2 11 3 4 10 7 6 8 5 9 12 55435 0 1 2 11 3 4 10 9 5 8 6 7 12 55483 0 1 2 11 3 5 8 6 7 9 4 10 12 55484 0 1 2 11 3 5 8 6 7 10 4 9 12 55517 0 1 2 11 3 5 9 4 10 7 6 8 12 55518 0 1 2 11 3 5 9 4 10 8 6 7 12 55591 0 1 2 11 3 6 7 5 8 9 4 10 12 55592 0 1 2 11 3 6 7 5 8 10 4 9 12 55601 0 1 2 11 3 6 7 8 5 9 4 10 12 55615 0 1 2 11 3 6 7 10 4 9 5 8 12 56813 0 1 2 11 3 10 4 5 9 7 6 8 12 56814 0 1 2 11 3 10 4 5 9 8 6 7 12 56815 0 1 2 11 3 10 4 6 7 8 5 9 12 56816 0 1 2 11 3 10 4 6 7 9 5 8 12 56855 0 1 2 11 3 10 4 9 5 8 6 7 12 56863 0 1 2 11 3 10 4 9 7 6 5 8 12 56865 0 1 2 11 3 10 4 9 8 5 6 7 12 57031 0 1 2 11 3 10 7 6 4 9 5 8 12 57033 0 1 2 11 3 10 7 6 5 8 4 9 12 57039 0 1 2 11 3 10 7 6 8 5 4 9 12 57079 0 1 2 11 3 10 8 5 7 6 4 9 12 57083 0 1 2 11 3 10 8 5 9 4 6 7 12 57111 0 1 2 11 3 10 9 4 5 8 6 7 12 57113 0 1 2 11 3 10 9 4 6 7 5 8 12 57123 0 1 2 11 3 10 9 4 8 5 6 7 12 64007 0 1 2 11 7 6 3 10 4 9 5 8 12 64015 0 1 2 11 7 6 3 10 8 5 4 9 12 64017 0 1 2 11 7 6 3 10 9 4 5 8 12 64047 0 1 2 11 7 6 4 9 8 5 3 10 12 64049 0 1 2 11 7 6 4 9 10 3 5 8 12 64073 0 1 2 11 7 6 5 8 4 9 3 10 12 64079 0 1 2 11 7 6 5 8 9 4 3 10 12 64081 0 1 2 11 7 6 5 8 10 3 4 9 12 64159 0 1 2 11 7 6 8 5 3 10 4 9 12 64161 0 1 2 11 7 6 8 5 4 9 3 10 12 64167 0 1 2 11 7 6 8 5 9 4 3 10 12 65203 0 1 2 11 8 5 3 10 7 6 4 9 12 65205 0 1 2 11 8 5 3 10 9 4 6 7 12 65235 0 1 2 11 8 5 4 9 7 6 3 10 12 65237 0 1 2 11 8 5 4 9 10 3 6 7 12 65267 0 1 2 11 8 5 6 7 9 4 3 10 12 65269 0 1 2 11 8 5 6 7 10 3 4 9 12 65347 0 1 2 11 8 5 7 6 3 10 4 9 12 65349 0 1 2 11 8 5 7 6 4 9 3 10 12 65423 0 1 2 11 8 5 9 4 10 3 6 7 12 66391 0 1 2 11 9 4 3 10 7 6 5 8 12 66393 0 1 2 11 9 4 3 10 8 5 6 7 12 66423 0 1 2 11 9 4 5 8 7 6 3 10 12 66425 0 1 2 11 9 4 5 8 10 3 6 7 12 66455 0 1 2 11 9 4 6 7 8 5 3 10 12 66457 0 1 2 11 9 4 6 7 10 3 5 8 12 66607 0 1 2 11 9 4 8 5 7 6 3 10 12 66663 0 1 2 11 9 4 10 3 5 8 6 7 12 66665 0 1 2 11 9 4 10 3 6 7 5 8 12 67571 0 1 2 11 10 3 4 9 5 8 6 7 12 67579 0 1 2 11 10 3 4 9 7 6 5 8 12 67581 0 1 2 11 10 3 4 9 8 5 6 7 12 67611 0 1 2 11 10 3 5 8 7 6 4 9 12 67613 0 1 2 11 10 3 5 8 9 4 6 7 12 67637 0 1 2 11 10 3 6 7 5 8 4 9 12 67643 0 1 2 11 10 3 6 7 8 5 4 9 12 67645 0 1 2 11 10 3 6 7 9 4 5 8 12 67851 0 1 2 11 10 3 9 4 5 8 6 7 12 67853 0 1 2 11 10 3 9 4 6 7 5 8 12 67863 0 1 2 11 10 3 9 4 8 5 6 7 12 98827 0 1 3 9 4 5 8 6 7 10 2 11 12 98828 0 1 3 9 4 5 8 6 7 11 2 10 12 98935 0 1 3 9 4 6 7 5 8 10 2 11 12 98936 0 1 3 9 4 6 7 5 8 11 2 10 12 99499 0 1 3 9 4 8 5 6 7 10 2 11 12 99500 0 1 3 9 4 8 5 6 7 11 2 10 12 112327 0 1 3 10 2 11 7 6 4 9 5 8 12 112329 0 1 3 10 2 11 7 6 5 8 4 9 12 112335 0 1 3 10 2 11 7 6 8 5 4 9 12 112375 0 1 3 10 2 11 8 5 7 6 4 9 12 112379 0 1 3 10 2 11 8 5 9 4 6 7 12 112407 0 1 3 10 2 11 9 4 5 8 6 7 12 112409 0 1 3 10 2 11 9 4 6 7 5 8 12 112419 0 1 3 10 2 11 9 4 8 5 6 7 12 112589 0 1 3 10 4 2 11 7 6 8 5 9 12 112603 0 1 3 10 4 2 11 9 5 8 6 7 12 112698 0 1 3 10 4 5 9 7 6 8 11 2 12 112706 0 1 3 10 4 5 9 8 6 7 11 2 12 112770 0 1 3 10 4 6 7 8 5 9 11 2 12 112778 0 1 3 10 4 6 7 9 5 8 11 2 12 113681 0 1 3 10 4 9 5 2 11 7 6 8 12 113682 0 1 3 10 4 9 5 2 11 8 6 7 12 113710 0 1 3 10 4 9 5 8 6 7 11 2 12 113715 0 1 3 10 4 9 5 8 7 6 2 11 12 113717 0 1 3 10 4 9 5 8 11 2 6 7 12 113842 0 1 3 10 4 9 7 6 5 8 11 2 12 113847 0 1 3 10 4 9 7 6 8 5 2 11 12 113882 0 1 3 10 4 9 8 5 6 7 11 2 12 113887 0 1 3 10 4 9 8 5 7 6 2 11 12 113919 0 1 3 10 4 9 11 2 5 8 6 7 12 113921 0 1 3 10 4 9 11 2 6 7 5 8 12 119311 0 1 3 10 7 6 2 11 8 5 4 9 12 119313 0 1 3 10 7 6 2 11 9 4 5 8 12 119338 0 1 3 10 7 6 4 9 5 8 11 2 12 119343 0 1 3 10 7 6 4 9 8 5 2 11 12 119345 0 1 3 10 7 6 4 9 11 2 5 8 12 119370 0 1 3 10 7 6 5 8 4 9 11 2 12 119375 0 1 3 10 7 6 5 8 9 4 2 11 12 119377 0 1 3 10 7 6 5 8 11 2 4 9 12 119458 0 1 3 10 7 6 8 5 4 9 11 2 12 119463 0 1 3 10 7 6 8 5 9 4 2 11 12 120499 0 1 3 10 8 5 2 11 7 6 4 9 12 120501 0 1 3 10 8 5 2 11 9 4 6 7 12 120531 0 1 3 10 8 5 4 9 7 6 2 11 12 120533 0 1 3 10 8 5 4 9 11 2 6 7 12 120563 0 1 3 10 8 5 6 7 9 4 2 11 12 120565 0 1 3 10 8 5 6 7 11 2 4 9 12 120646 0 1 3 10 8 5 7 6 4 9 11 2 12 120710 0 1 3 10 8 5 9 4 6 7 11 2 12 121687 0 1 3 10 9 4 2 11 7 6 5 8 12 121689 0 1 3 10 9 4 2 11 8 5 6 7 12 121714 0 1 3 10 9 4 5 8 6 7 11 2 12 121719 0 1 3 10 9 4 5 8 7 6 2 11 12 121721 0 1 3 10 9 4 5 8 11 2 6 7 12 121746 0 1 3 10 9 4 6 7 5 8 11 2 12 121751 0 1 3 10 9 4 6 7 8 5 2 11 12 121753 0 1 3 10 9 4 6 7 11 2 5 8 12 121898 0 1 3 10 9 4 8 5 6 7 11 2 12 121903 0 1 3 10 9 4 8 5 7 6 2 11 12 122867 0 1 3 10 11 2 4 9 5 8 6 7 12 122875 0 1 3 10 11 2 4 9 7 6 5 8 12 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5 2 12 135852 0 1 3 11 9 4 10 7 6 8 5 2 12 135860 0 1 3 11 9 4 10 8 5 7 6 2 12 136146 0 1 3 11 9 5 8 6 7 10 4 2 12 137404 0 1 3 11 10 4 9 5 8 7 6 2 12 137436 0 1 3 11 10 4 9 7 6 8 5 2 12 137444 0 1 3 11 10 4 9 8 5 7 6 2 12 156113 0 1 4 8 5 6 7 9 3 10 2 11 12 156114 0 1 4 8 5 6 7 9 3 10 11 2 12 156122 0 1 4 8 5 6 7 10 3 9 11 2 12 156127 0 1 4 8 5 6 7 11 2 10 3 9 12 156365 0 1 4 8 5 7 6 2 11 9 3 10 12 156366 0 1 4 8 5 7 6 2 11 10 3 9 12 156367 0 1 4 8 5 7 6 3 9 10 2 11 12 156368 0 1 4 8 5 7 6 3 9 11 2 10 12 167947 0 1 4 9 3 5 8 6 7 10 2 11 12 167948 0 1 4 9 3 5 8 6 7 11 2 10 12 168055 0 1 4 9 3 6 7 5 8 10 2 11 12 168056 0 1 4 9 3 6 7 5 8 11 2 10 12 168967 0 1 4 9 3 10 2 11 7 6 5 8 12 168969 0 1 4 9 3 10 2 11 8 5 6 7 12 169138 0 1 4 9 3 10 7 6 5 8 11 2 12 169143 0 1 4 9 3 10 7 6 8 5 2 11 12 169178 0 1 4 9 3 10 8 5 6 7 11 2 12 169183 0 1 4 9 3 10 8 5 7 6 2 11 12 169215 0 1 4 9 3 10 11 2 5 8 6 7 12 169217 0 1 4 9 3 10 11 2 6 7 5 8 12 169745 0 1 4 9 5 2 11 3 10 7 6 8 12 169746 0 1 4 9 5 2 11 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8 4 5 7 6 2 10 3 9 12 883413 0 1 11 8 4 5 7 6 3 9 2 10 12 883663 0 1 11 8 4 6 7 5 2 10 3 9 12 883665 0 1 11 8 4 6 7 5 3 9 2 10 12 884235 0 1 11 8 4 9 3 5 7 6 2 10 12 884247 0 1 11 8 4 9 3 6 7 5 2 10 12 884919 0 1 11 8 5 2 10 3 9 4 6 7 12 885359 0 1 11 8 5 4 9 3 10 2 6 7 12 885931 0 1 11 8 5 7 6 2 10 3 4 9 12 885939 0 1 11 8 5 7 6 3 9 4 2 10 12 885943 0 1 11 8 5 7 6 3 10 2 4 9 12 885951 0 1 11 8 5 7 6 4 9 3 2 10 12 886159 0 1 11 8 5 9 4 2 10 3 6 7 12 886171 0 1 11 8 5 9 4 3 10 2 6 7 12 890719 0 1 11 9 3 4 8 5 7 6 2 10 12 891159 0 1 11 9 3 6 7 5 8 4 2 10 12 891723 0 1 11 9 3 8 4 5 7 6 2 10 12 891735 0 1 11 9 3 8 4 6 7 5 2 10 12 891943 0 1 11 9 3 10 2 4 8 5 6 7 12 891951 0 1 11 9 3 10 2 5 7 6 4 8 12 891955 0 1 11 9 3 10 2 5 8 4 6 7 12 891963 0 1 11 9 3 10 2 6 7 5 4 8 12 892395 0 1 11 9 4 2 10 3 5 8 6 7 12 892397 0 1 11 9 4 2 10 3 6 7 5 8 12 892647 0 1 11 9 4 3 10 2 5 8 6 7 12 892649 0 1 11 9 4 3 10 2 6 7 5 8 12 893647 0 1 11 9 4 8 5 2 10 3 6 7 12 893659 0 1 11 9 4 8 5 3 10 2 6 7 12 898011 0 1 11 10 2 3 9 4 5 8 6 7 12 898013 0 1 11 10 2 3 9 4 6 7 5 8 12 898023 0 1 11 10 2 3 9 4 8 5 6 7 12 898207 0 1 11 10 2 4 8 5 7 6 3 9 12 898263 0 1 11 10 2 4 9 3 5 8 6 7 12 898265 0 1 11 10 2 4 9 3 6 7 5 8 12 898387 0 1 11 10 2 5 7 6 3 9 4 8 12 898389 0 1 11 10 2 5 7 6 4 8 3 9 12 898463 0 1 11 10 2 5 8 4 9 3 6 7 12 898639 0 1 11 10 2 6 7 5 3 9 4 8 12 898641 0 1 11 10 2 6 7 5 4 8 3 9 12 898647 0 1 11 10 2 6 7 5 8 4 3 9 12 899431 0 1 11 10 2 9 3 4 8 5 6 7 12 899439 0 1 11 10 2 9 3 5 7 6 4 8 12 899443 0 1 11 10 2 9 3 5 8 4 6 7 12 899451 0 1 11 10 2 9 3 6 7 5 4 8 12 L= 907200 Crossings, # 0 616 =2*A079410(6) 1 2612 2 6578 3 13258 4 22951 5 34551 6 47558 7 60164 8 69805 9 76233 10 78956 11 75608 12 68885 13 61388 14 52725 15 45361 16 38418 17 31105 18 27315 19 23763 20 18121 21 14565 22 11554 23 8148 24 5697 25 3467 26 2443 27 1756 28 969 29 554 30 513 31 523 32 290 33 154 34 214 35 183 36 60 37 39 38 52 39 32 41 7 42 8 45 1 (45=A000384(5)) -------------------------------------------------------------------- N = 7 L= 79315200 Crossings, # 0 5780 =2*A079410(7) 1 28062 2 83830 3 192946 4 378774 5 652995 6 1025455 7 1488574 8 2024878 9 2597909 10 3185399 11 3742655 12 4231953 13 4637975 14 4941540 15 5104362 16 5148296 17 5042115 18 4789037 19 4453230 20 4025845 21 3536097 22 3056490 23 2599872 24 2182056 25 1829128 26 1504727 27 1284212 28 1069388 29 915349 30 752362 31 632230 32 525210 33 417623 34 323570 35 246352 36 189223 37 136818 38 93125 39 65577 40 48613 41 34123 42 21734 43 15337 44 13916 45 10476 46 6528 47 5303 48 5647 49 3804 50 2168 51 1898 52 1858 53 1048 54 467 55 492 56 408 57 142 58 62 59 86 60 51 62 9 63 10 66 1 (66=A000384(6)) ------------------------------------------------------------------- N = 8 L= 9551001600 Crossings, # 0 51528 =2*A079410(8) 1 299128 2 1045172 3 2780302 4 6241794 5 12310016 6 22018027 7 36369449 8 56245840 9 82114752 10 114073821 11 151498103 12 193255370 13 237674310 14 282768258 15 326264220 16 366219004 17 400886775 18 428991734 19 449812265 20 463375382 21 469945084 22 469948095 23 464068336 24 452851598 25 437142549 26 416958503 27 392759346 28 365523227 29 335717866 30 303915875 31 271198770 32 238486590 33 207406633 34 177902473 35 151055613 36 127087332 37 106812272 38 89629695 39 74832580 40 62740058 41 53131645 42 44893184 43 37717620 44 31840339 45 26722499 46 22410053 47 18268639 48 14828064 49 11948620 50 9504110 51 7287812 52 5537479 53 4238549 54 3194655 55 2324204 56 1662612 57 1259014 58 959752 59 683270 60 489526 61 400773 62 333114 63 240187 64 176184 65 159831 66 134829 67 92655 68 67071 69 62041 70 48732 71 29986 72 21164 73 19436 74 13486 75 7200 76 5028 77 4596 78 2596 79 1054 80 910 81 753 82 268 83 89 84 128 85 74 87 11 88 12 91 1 (91=A000384(7))