Av(12345)
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Counting Sequence
1, 1, 2, 6, 24, 119, 694, 4582, 33324, 261808, 2190688, 19318688, 178108704, 1705985883, 16891621166, ...

This specification was found using the strategy pack "Row Placements Tracked Fusion Expand Verified" and has 30 rules.

Found on January 22, 2022.

Finding the specification took 27 seconds.

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Created with Raphaël 2.1.4
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Copy 30 equations to clipboard:
F0(x)=F1(x)+F2(x)F1(x)=1F2(x)=F23(x)F3(x)F3(x)=F4(x,1)F4(x,y0)=F1(x)+F28(x,y0)+F5(x,y0)F5(x,y0)=F14(x,y0)F6(x,y0)F6(x,y0)=F7(x,1,y0)F7(x,y0,y1)=F8(x,y0y1,y1)F8(x,y0,y1)=F1(x)+F24(x,y0,y1)+F26(x,y0,y1)+F9(x,y0,y1)F9(x,y0,y1)=F10(x,y0,y1)F14(x,y0)F10(x,y0,y1)=F11(x,y0,1,y1)F11(x,y0,y1,y2)=F12(x,y0,y0y1,y2)F12(x,y0,y1,y2)=F1(x)+F13(x,y0,y1,y2)+F15(x,y0,y1,y2)+F18(x,y0,y1,y2)+F21(x,y0,y1,y2)F13(x,y0,y1,y2)=F12(x,y0,y1,y2)F14(x,y0)F14(x,y0)=y0xF15(x,y0,y1,y2)=F14(x,y1)F16(x,y0,y1,y2)F16(x,y0,y1,y2)=y1F17(x,1,y1,y2)+y0F17(x,y0y1,y1,y2)y1+y0F17(x,y0,y1,y2)=F12(x,y0y1,y1,y2)F18(x,y0,y1,y2)=F14(x,y2)F19(x,y0,y1,y2)F19(x,y0,y1,y2)=y2F20(x,y0,1,y2)+y1F20(x,y0,y1y2,y2)y2+y1F20(x,y0,y1,y2)=F12(x,y0,y1y2,y2)F21(x,y0,y1,y2)=F22(x,y0,y1,y2)F23(x)F22(x,y0,y1,y2)=y2F12(x,y0,y1,y2)F12(x,y0,y1,1)1+y2F23(x)=xF24(x,y0,y1)=F14(x,y1)F25(x,y0,y1)F25(x,y0,y1)=y1F7(x,1,y1)+y0F7(x,y0y1,y1)y1+y0F26(x,y0,y1)=F23(x)F27(x,y0,y1)F27(x,y0,y1)=y1F8(x,y0,y1)F8(x,y0,1)1+y1F28(x,y0)=F23(x)F29(x,y0)F29(x,y0)=y0F4(x,y0)F4(x,1)1+y0

This specification was found using the strategy pack "Row Placements Tracked Fusion Req Corrob Expand Verified" and has 38 rules.

Found on November 19, 2021.

Finding the specification took 174 seconds.

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Created with Raphaël 2.1.4
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Copy 38 equations to clipboard:
F0(x)=F1(x)+F2(x)F1(x)=1F2(x)=F27(x)F3(x)F3(x)=F1(x)+F32(x)+F4(x)F4(x)=F27(x)F5(x)F5(x)=F6(x,1)F6(x,y0)=F7(x,1,y0)F7(x,y0,y1)=F8(x,y0y1,y1)F8(x,y0,y1)=F1(x)+F28(x,y0,y1)+F30(x,y0,y1)+F9(x,y0,y1)F9(x,y0,y1)=F10(x,y0,y1)F14(x,y0)F10(x,y0,y1)=F11(x,1,y0,y1)F11(x,y0,y1,y2)=F12(x,y0y1,y1,y2)F12(x,y0,y1,y2)=F1(x)+F13(x,y0,y1,y2)+F15(x,y0,y1,y2)+F17(x,y0,y1,y2)+F25(x,y0,y1,y2)F13(x,y0,y1,y2)=F12(x,y0,y1,y2)F14(x,y0)F14(x,y0)=y0xF15(x,y0,y1,y2)=F14(x,y1)F16(x,y0,y1,y2)F16(x,y0,y1,y2)=y1F11(x,1,y1,y2)y0F11(x,y0y1,y1,y2)y1+y0F17(x,y0,y1,y2)=F14(x,y2)F18(x,y0,y1,y2)F18(x,y0,y1,y2)=F19(x,y0,y1,1,y2)F19(x,y0,y1,y2,y3)=F20(x,y0,1,y1,y2,y3)F20(x,y0,y1,y2,y3,y4)=y1y2F21(x,y0,y1,y2y4,y3y4)+y4F21(x,y0,y1,1y1,y3y4)y1y2y4F21(x,y0,y1,y2,y3)=F22(x,y0,y1,1,y2,y3)F22(x,y0,y1,y2,y3,y4)=F23(x,y0y1,y2,y1y3,y4)F23(x,y0,y1,y2,y3)=F24(x,y0y1,y2,y1y3)F24(x,y0,y1,y2)=F12(x,y0,y1y2,y2)F25(x,y0,y1,y2)=F26(x,y0,y1,y2)F27(x)F26(x,y0,y1,y2)=y2F12(x,y0,y1,y2)+F12(x,y0,y1,1)1+y2F27(x)=xF28(x,y0,y1)=F14(x,y1)F29(x,y0,y1)F29(x,y0,y1)=y1F7(x,1,y1)y0F7(x,y0y1,y1)y1+y0F30(x,y0,y1)=F27(x)F31(x,y0,y1)F31(x,y0,y1)=y1F8(x,y0,y1)+F8(x,y0,1)1+y1F32(x)=F27(x)F33(x)F33(x)=F34(x,1)F34(x,y0)=y0F35(x,y0)+F35(x,1)1+y0F35(x,y0)=F1(x)+F36(x,y0)+F37(x,y0)F36(x,y0)=F14(x,y0)F6(x,y0)F37(x,y0)=F27(x)F34(x,y0)

This specification was found using the strategy pack "Point And Row Placements Tracked Fusion Expand Verified" and has 34 rules.

Found on November 19, 2021.

Finding the specification took 92 seconds.

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Created with Raphaël 2.1.4
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Copy 34 equations to clipboard:
F0(x)=F1(x)+F2(x)F1(x)=1F2(x)=F3(x)F3(x)=F23(x)F4(x)F4(x)=F1(x)+F28(x)+F5(x)F5(x)=F23(x)F6(x)F6(x)=F7(x,1)F7(x,y0)=F8(x,1,y0)F8(x,y0,y1)=F9(x,y0y1,y1)F9(x,y0,y1)=F1(x)+F10(x,y0,y1)+F24(x,y0,y1)+F26(x,y0,y1)F10(x,y0,y1)=F11(x,y0,y1)F15(x,y0)F11(x,y0,y1)=F12(x,1,y0,y1)F12(x,y0,y1,y2)=F13(x,y0y1,y1,y2)F13(x,y0,y1,y2)=F1(x)+F14(x,y0,y1,y2)+F16(x,y0,y1,y2)+F18(x,y0,y1,y2)+F21(x,y0,y1,y2)F14(x,y0,y1,y2)=F13(x,y0,y1,y2)F15(x,y0)F15(x,y0)=y0xF16(x,y0,y1,y2)=F15(x,y1)F17(x,y0,y1,y2)F17(x,y0,y1,y2)=y1F12(x,1,y1,y2)y0F12(x,y0y1,y1,y2)y1+y0F18(x,y0,y1,y2)=F15(x,y2)F19(x,y0,y1,y2)F19(x,y0,y1,y2)=y2F20(x,y0,1,y2)y1F20(x,y0,y1y2,y2)y2+y1F20(x,y0,y1,y2)=F13(x,y0,y1y2,y2)F21(x,y0,y1,y2)=F22(x,y0,y1,y2)F23(x)F22(x,y0,y1,y2)=y2F13(x,y0,y1,y2)+F13(x,y0,y1,1)1+y2F23(x)=xF24(x,y0,y1)=F15(x,y1)F25(x,y0,y1)F25(x,y0,y1)=y1F8(x,1,y1)y0F8(x,y0y1,y1)y1+y0F26(x,y0,y1)=F23(x)F27(x,y0,y1)F27(x,y0,y1)=y1F9(x,y0,y1)+F9(x,y0,1)1+y1F28(x)=F23(x)F29(x)F29(x)=F30(x,1)F30(x,y0)=y0F31(x,y0)+F31(x,1)1+y0F31(x,y0)=F1(x)+F32(x,y0)+F33(x,y0)F32(x,y0)=F15(x,y0)F7(x,y0)F33(x,y0)=F23(x)F30(x,y0)

This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 37 rules.

Found on June 09, 2021.

Finding the specification took 185 seconds.

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Created with Raphaël 2.1.4
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Copy 37 equations to clipboard:
F0(x)=F1(x)+F2(x)F1(x)=1F2(x)=F21(x)F3(x)F3(x)=F1(x)+F27(x)+F4(x)F4(x)=F21(x)F5(x)F5(x)=F6(x,1)F6(x,y0)=F7(x,y0,1)F7(x,y0,y1)=F1(x)+F22(x,y0,y1)+F25(x,y0,y1)+F8(x,y0,y1)F8(x,y0,y1)=F13(x,y0)F9(x,y0,y1)F9(x,y0,y1)=F10(x,1,y0,y1)F10(x,y0,y1,y2)=F11(x,y0y1,y1,y2)F11(x,y0,y1,y2)=F1(x)+F12(x,y0,y1,y2)+F14(x,y0,y1,y2)+F16(x,y0,y1,y2)+F19(x,y0,y1,y2)F12(x,y0,y1,y2)=F11(x,y0,y1,y2)F13(x,y0)F13(x,y0)=y0xF14(x,y0,y1,y2)=F13(x,y1)F15(x,y0,y1,y2)F15(x,y0,y1,y2)=y1F10(x,1,y1,y2)y0F10(x,y0y1,y1,y2)y1+y0F16(x,y0,y1,y2)=F13(x,y2)F17(x,y0,y1,y2)F17(x,y0,y1,y2)=y2F18(x,y0,1,y2)y1F18(x,y0,y1y2,y2)y2+y1F18(x,y0,y1,y2)=F11(x,y0,y1y2,y2)F19(x,y0,y1,y2)=F20(x,y0,y1,y2)F21(x)F20(x,y0,y1,y2)=y2F11(x,y0,y1,y2)+F11(x,y0,y1,1)1+y2F21(x)=xF22(x,y0,y1)=F13(x,y1)F23(x,y0,y1)F23(x,y0,y1)=y1F24(x,1,y1)y0F24(x,y0y1,y1)y1+y0F24(x,y0,y1)=F7(x,y0y1,y1)F25(x,y0,y1)=F21(x)F26(x,y0,y1)F26(x,y0,y1)=y1F7(x,y0,y1)+F7(x,y0,1)1+y1F27(x)=F21(x)F28(x)F28(x)=F29(x,1)F29(x,y0)=F1(x)+F30(x,y0)+F33(x,y0)+F35(x,y0)F30(x,y0)=F13(x,y0)F31(x,y0)F31(x,y0)=y0F32(x,y0,1)F32(x,y0,1y0)1+y0F32(x,y0,y1)=F7(x,y0,y0y1)F33(x,y0)=F21(x)F34(x,y0)F34(x,y0)=y0F6(x,y0)+F6(x,1)1+y0F35(x,y0)=F21(x)F36(x,y0)F36(x,y0)=y0F29(x,y0)+F29(x,1)1+y0