Av(1342, 3412)
View Raw Data
Generating Function
(25x26x+1x4x25x26x+1+3x)(x1)4x420x3+24x210x+1
Counting Sequence
1, 1, 2, 6, 22, 88, 366, 1556, 6720, 29396, 129996, 580276, 2611290, 11834116, 53963190, ...
Implicit Equation for the Generating Function
(4x420x3+24x210x+1)F(x)2+2x(4x3)(x1)F(x)(x1)2=0
Recurrence
a(0)=1
a(1)=1
a(2)=2
a(3)=6
a(4)=22
a(5)=88
a(6)=366
a(n+7)=40(n+1)a(n)n+74(105+67n)a(n+1)n+7+4(377+153n)a(n+2)n+716(143+42n)a(n+3)n+7+28(61+14n)a(n+4)n+7(643+121n)a(n+5)n+7+2(56+9n)a(n+6)n+7,n7
Heatmap

To create this heatmap, we sampled 1,000,000 permutations of length 300 uniformly at random. The color of the point (i,j) represents how many permutations have value j at index i (darker = more).

This specification was found using the strategy pack "Point And Col Placements" and has 26 rules.

Found on January 17, 2022.

Finding the specification took 98 seconds.

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Created with Raphaël 2.1.4
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Copy 26 equations to clipboard:
F0(x)=F1(x)+F2(x)F1(x)=1F2(x)=F3(x)F4(x)F3(x)=xF4(x)=F0(x)+F5(x)F5(x)=F6(x)F6(x)=F3(x)F7(x)F7(x)=F24(x)+F8(x)F8(x)=F0(x)+F6(x)+F9(x)F9(x)=F10(x)F10(x)=F11(x)F3(x)F11(x)=F12(x)+F21(x)F12(x)=F0(x)+F13(x)+F9(x)F13(x)=F14(x)F14(x)=F0(x)F15(x)F3(x)F15(x)=F16(x)+F19(x)F16(x)=F1(x)+F17(x)+F18(x)F17(x)=F16(x)F3(x)F18(x)=F15(x)F3(x)F19(x)=F20(x)F20(x)=F15(x)F16(x)F3(x)F21(x)=F22(x)F3(x)F22(x)=F11(x)+F23(x)F23(x)=F16(x)F5(x)F24(x)=F25(x)F25(x)=F16(x)F3(x)F7(x)

This specification was found using the strategy pack "Row And Col Placements" and has 54 rules.

Found on January 18, 2022.

Finding the specification took 34 seconds.

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Created with Raphaël 2.1.4
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Copy 54 equations to clipboard:
F0(x)=F1(x)+F2(x)F1(x)=1F2(x)=F3(x)F8(x)F3(x)=F1(x)+F4(x)+F42(x)F4(x)=F5(x)F8(x)F5(x)=F6(x)+F9(x)F6(x)=F1(x)+F4(x)+F7(x)F7(x)=F6(x)F8(x)F8(x)=xF9(x)=F10(x)F10(x)=F11(x)F23(x)F8(x)F11(x)=F12(x)+F36(x)+F38(x)F12(x)=F1(x)+F13(x)+F32(x)F13(x)=F14(x)F8(x)F14(x)=F15(x)+F17(x)F15(x)=F16(x)F8(x)F16(x)=F1(x)+F0(x)F17(x)=F18(x)F18(x)=F19(x)F8(x)F19(x)=F11(x)+F20(x)F20(x)=F21(x)F21(x)=F15(x)F22(x)F23(x)F8(x)F22(x)=F23(x)+F26(x)F23(x)=F1(x)+F24(x)F24(x)=F25(x)F25(x)=F23(x)F8(x)F26(x)=F24(x)+F27(x)F27(x)=F28(x)+F29(x)+F31(x)F28(x)=0F29(x)=F30(x)F8(x)F30(x)=F24(x)+F27(x)F31(x)=F26(x)F8(x)F32(x)=F33(x)F8(x)F33(x)=F34(x)F8(x)F34(x)=F1(x)F35(x)+F15(x)F35(x)=F14(x)F8(x)F36(x)=F37(x)F37(x)=F19(x)F8(x)F38(x)=F39(x)F8(x)F39(x)=F40(x)F8(x)F40(x)=F41(x)F41(x)=F12(x)+F33(x)F42(x)=F43(x)F8(x)F43(x)=F3(x)+F44(x)F44(x)=F45(x)F45(x)=F46(x)F8(x)F46(x)=F11(x)+F47(x)+F52(x)F47(x)=F48(x)F48(x)=F19(x)F49(x)F8(x)F49(x)=F23(x)+F50(x)F50(x)=F51(x)F51(x)=F49(x)2F8(x)F52(x)=F53(x)F53(x)=F39(x)F49(x)F8(x)

This specification was found using the strategy pack "Point And Row Placements" and has 59 rules.

Found on January 18, 2022.

Finding the specification took 39 seconds.

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Created with Raphaël 2.1.4
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Copy 59 equations to clipboard:
F0(x)=F1(x)+F2(x)F1(x)=1F2(x)=F3(x)F3(x)=F13(x)F4(x)F4(x)=F0(x)+F5(x)F5(x)=F6(x)F6(x)=F13(x)F7(x)F7(x)=F23(x)+F8(x)F8(x)=F0(x)+F22(x)+F9(x)F9(x)=F10(x)F13(x)F14(x)F10(x)=F1(x)+F11(x)F11(x)=F12(x)F12(x)=F10(x)F13(x)F13(x)=xF14(x)=F15(x)+F18(x)F15(x)=F1(x)+F16(x)+F17(x)F16(x)=F13(x)F14(x)F17(x)=F13(x)F15(x)F18(x)=F19(x)+F20(x)F19(x)=F0(x)+F4(x)F20(x)=F21(x)F21(x)=F13(x)F18(x)F22(x)=F5(x)F23(x)=F24(x)F24(x)=F13(x)F25(x)F25(x)=F26(x)+F56(x)F26(x)=F50(x)+F27(x)F27(x)=F28(x)+F30(x)+F48(x)F28(x)=F29(x)F13(x)F29(x)=F19(x)F30(x)=F31(x)F31(x)=F13(x)F32(x)F45(x)F32(x)=F33(x)F13(x)F33(x)=F34(x)F34(x)=F4(x)F43(x)+F35(x)F35(x)=F36(x)F13(x)F36(x)=F37(x)F37(x)=F40(x)+F38(x)F38(x)=F39(x)F13(x)F39(x)=F2(x)F40(x)=F1(x)+F41(x)F41(x)=F42(x)F42(x)=F40(x)2F13(x)F43(x)=F44(x)F44(x)=F13(x)F27(x)F45(x)=F10(x)+F46(x)F46(x)=F47(x)F47(x)=F45(x)2F13(x)F48(x)=F49(x)F49(x)=F13(x)F27(x)F45(x)F50(x)=F51(x)F51(x)=F45(x)F52(x)F52(x)=F53(x)+F35(x)F53(x)=F4(x)+F54(x)F54(x)=F55(x)F55(x)=F13(x)F26(x)F56(x)=F57(x)F57(x)=F45(x)F58(x)F58(x)=F53(x)+F7(x)

This specification was found using the strategy pack "Insertion Row And Col Placements Req Corrob" and has 77 rules.

Found on January 17, 2022.

Finding the specification took 174 seconds.

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Copy 77 equations to clipboard:
F0(x)=F1(x)+F2(x)F1(x)=1F2(x)=F3(x)F3(x)=F4(x)F5(x)F4(x)=xF5(x)=F6(x)+F9(x)F6(x)=F1(x)+F7(x)F7(x)=F8(x)F8(x)=F6(x)2F4(x)F9(x)=F10(x)+F2(x)F10(x)=F11(x)F11(x)=F12(x)F4(x)F12(x)=F13(x)+F25(x)F13(x)=F14(x)F6(x)F14(x)=F15(x)F15(x)=F16(x)F4(x)F16(x)=F17(x)+F20(x)+F22(x)F17(x)=F1(x)+F18(x)F18(x)=F19(x)F19(x)=F17(x)F4(x)F20(x)=F21(x)F21(x)=F16(x)F4(x)F6(x)F22(x)=F23(x)F23(x)=F16(x)F24(x)F4(x)F24(x)=F15(x)+F6(x)F25(x)=F26(x)+F69(x)F26(x)=F27(x)+F28(x)F27(x)=F2(x)F6(x)F28(x)=F29(x)+F42(x)F29(x)=F30(x)F30(x)=F31(x)F4(x)F31(x)=F32(x)+F33(x)F32(x)=F17(x)F2(x)F33(x)=F34(x)+F35(x)F34(x)=F2(x)+F29(x)F35(x)=F2(x)F36(x)F36(x)=F18(x)+F37(x)F37(x)=F38(x)+F39(x)+F41(x)F38(x)=0F39(x)=F4(x)F40(x)F40(x)=F18(x)+F37(x)F41(x)=F36(x)F4(x)F42(x)=F38(x)+F43(x)+F60(x)F43(x)=F44(x)F44(x)=F4(x)F45(x)F45(x)=F46(x)+F54(x)F46(x)=F47(x)+F48(x)F47(x)=F2(x)F7(x)F48(x)=F49(x)F49(x)=F4(x)F50(x)F50(x)=F51(x)+F52(x)F51(x)=F14(x)F2(x)F6(x)F52(x)=F24(x)F53(x)F53(x)=F29(x)+F48(x)F54(x)=F55(x)+F59(x)F55(x)=F2(x)F56(x)F56(x)=F57(x)+F7(x)F57(x)=F4(x)F58(x)F58(x)=F20(x)+F22(x)+F38(x)F59(x)=F18(x)F46(x)F60(x)=F4(x)F61(x)F61(x)=F53(x)+F62(x)F62(x)=F42(x)+F63(x)F63(x)=F64(x)F64(x)=F4(x)F65(x)F65(x)=F66(x)+F68(x)F66(x)=F67(x)F67(x)=F14(x)F2(x)F6(x)F7(x)F68(x)=F24(x)F62(x)F69(x)=F70(x)+F71(x)F70(x)=F10(x)F6(x)F71(x)=F72(x)+F74(x)F72(x)=F73(x)F73(x)=F12(x)F18(x)F4(x)F74(x)=F75(x)F75(x)=F12(x)F4(x)F76(x)F76(x)=F57(x)

This specification was found using the strategy pack "Insertion Point Placements Req Corrob" and has 90 rules.

Found on January 17, 2022.

Finding the specification took 147 seconds.

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Copy 90 equations to clipboard:
F0(x)=F1(x)+F2(x)F1(x)=1F2(x)=F3(x)F3(x)=F4(x)F5(x)F4(x)=xF5(x)=F6(x)+F9(x)F6(x)=F1(x)+F7(x)F7(x)=F8(x)F8(x)=F6(x)2F4(x)F9(x)=F10(x)+F2(x)F10(x)=F11(x)F11(x)=F12(x)F4(x)F12(x)=F13(x)+F27(x)F13(x)=F14(x)F6(x)F14(x)=F15(x)F15(x)=F16(x)F4(x)F16(x)=F17(x)+F20(x)+F22(x)F17(x)=F1(x)+F18(x)F18(x)=F19(x)F19(x)=F17(x)F4(x)F20(x)=F21(x)F21(x)=F16(x)F4(x)F6(x)F22(x)=F23(x)F23(x)=F16(x)F24(x)F4(x)F24(x)=F17(x)+F25(x)F25(x)=F26(x)F26(x)=F24(x)2F4(x)F27(x)=F28(x)+F85(x)F28(x)=F29(x)+F30(x)F29(x)=F2(x)F6(x)F30(x)=F31(x)+F38(x)F31(x)=F32(x)F32(x)=F33(x)F4(x)F33(x)=F34(x)+F35(x)F34(x)=F2(x)+F31(x)F35(x)=F36(x)+F37(x)F36(x)=F17(x)F2(x)F37(x)=F18(x)F34(x)F38(x)=F39(x)F39(x)=F4(x)F40(x)F40(x)=F41(x)+F63(x)F41(x)=F42(x)F6(x)F42(x)=F43(x)+F48(x)F43(x)=F44(x)F44(x)=F4(x)F45(x)F45(x)=F46(x)+F47(x)F46(x)=F18(x)+F43(x)F47(x)=F46(x)F6(x)F48(x)=F49(x)+F53(x)F49(x)=F50(x)F50(x)=F4(x)F51(x)F51(x)=F20(x)+F22(x)+F52(x)F52(x)=0F53(x)=F54(x)F54(x)=F4(x)F55(x)F55(x)=F56(x)+F62(x)F56(x)=F57(x)+F61(x)F57(x)=F53(x)+F58(x)F58(x)=F59(x)F59(x)=F4(x)F60(x)F60(x)=F18(x)+F58(x)F61(x)=F14(x)F46(x)F62(x)=F24(x)F57(x)F63(x)=F64(x)+F78(x)F64(x)=F65(x)+F66(x)F65(x)=F2(x)F43(x)F66(x)=F38(x)+F67(x)F67(x)=F68(x)F68(x)=F4(x)F69(x)F69(x)=F70(x)+F76(x)F70(x)=F71(x)+F74(x)F71(x)=F67(x)+F72(x)F72(x)=F73(x)F73(x)=F18(x)F31(x)F74(x)=F75(x)F75(x)=F14(x)F2(x)F46(x)F76(x)=F77(x)F77(x)=F24(x)F71(x)F78(x)=F79(x)+F80(x)F79(x)=F10(x)F43(x)F80(x)=F81(x)+F83(x)F81(x)=F82(x)F82(x)=F12(x)F4(x)F49(x)F83(x)=F84(x)F84(x)=F12(x)F4(x)F53(x)F85(x)=F86(x)+F87(x)F86(x)=F10(x)F6(x)F87(x)=F81(x)+F88(x)F88(x)=F89(x)F89(x)=F10(x)F17(x)F4(x)