Identifier
Values
=>
[1]=>1 [1,1]=>1 [2]=>1 [1,1,1]=>1 [1,2]=>2 [2,1]=>2 [3]=>1 [1,1,1,1]=>1 [1,1,2]=>3 [1,2,1]=>5 [1,3]=>3 [2,1,1]=>3 [2,2]=>5 [3,1]=>3 [4]=>1 [1,1,1,1,1]=>1 [1,1,1,2]=>4 [1,1,2,1]=>9 [1,1,3]=>6 [1,2,1,1]=>9 [1,2,2]=>16 [1,3,1]=>11 [1,4]=>4 [2,1,1,1]=>4 [2,1,2]=>11 [2,2,1]=>16 [2,3]=>9 [3,1,1]=>6 [3,2]=>9 [4,1]=>4 [5]=>1 [1,1,1,1,1,1]=>1 [1,1,1,1,2]=>5 [1,1,1,2,1]=>14 [1,1,1,3]=>10 [1,1,2,1,1]=>19 [1,1,2,2]=>35 [1,1,3,1]=>26 [1,1,4]=>10 [1,2,1,1,1]=>14 [1,2,1,2]=>40 [1,2,2,1]=>61 [1,2,3]=>35 [1,3,1,1]=>26 [1,3,2]=>40 [1,4,1]=>19 [1,5]=>5 [2,1,1,1,1]=>5 [2,1,1,2]=>19 [2,1,2,1]=>40 [2,1,3]=>26 [2,2,1,1]=>35 [2,2,2]=>61 [2,3,1]=>40 [2,4]=>14 [3,1,1,1]=>10 [3,1,2]=>26 [3,2,1]=>35 [3,3]=>19 [4,1,1]=>10 [4,2]=>14 [5,1]=>5 [6]=>1 [1,1,1,1,1,1,1]=>1 [1,1,1,1,1,2]=>6 [1,1,1,1,2,1]=>20 [1,1,1,1,3]=>15 [1,1,1,2,1,1]=>34 [1,1,1,2,2]=>64 [1,1,1,3,1]=>50 [1,1,1,4]=>20 [1,1,2,1,1,1]=>34 [1,1,2,1,2]=>99 [1,1,2,2,1]=>155 [1,1,2,3]=>90 [1,1,3,1,1]=>71 [1,1,3,2]=>111 [1,1,4,1]=>55 [1,1,5]=>15 [1,2,1,1,1,1]=>20 [1,2,1,1,2]=>78 [1,2,1,2,1]=>169 [1,2,1,3]=>111 [1,2,2,1,1]=>155 [1,2,2,2]=>272 [1,2,3,1]=>181 [1,2,4]=>64 [1,3,1,1,1]=>50 [1,3,1,2]=>132 [1,3,2,1]=>181 [1,3,3]=>99 [1,4,1,1]=>55 [1,4,2]=>78 [1,5,1]=>29 [1,6]=>6 [2,1,1,1,1,1]=>6 [2,1,1,1,2]=>29 [2,1,1,2,1]=>78 [2,1,1,3]=>55 [2,1,2,1,1]=>99 [2,1,2,2]=>181 [2,1,3,1]=>132 [2,1,4]=>50 [2,2,1,1,1]=>64 [2,2,1,2]=>181 [2,2,2,1]=>272 [2,2,3]=>155 [2,3,1,1]=>111 [2,3,2]=>169 [2,4,1]=>78 [2,5]=>20 [3,1,1,1,1]=>15 [3,1,1,2]=>55 [3,1,2,1]=>111 [3,1,3]=>71 [3,2,1,1]=>90 [3,2,2]=>155 [3,3,1]=>99 [3,4]=>34 [4,1,1,1]=>20 [4,1,2]=>50 [4,2,1]=>64 [4,3]=>34 [5,1,1]=>15 [5,2]=>20 [6,1]=>6 [7]=>1 [1,1,1,1,1,1,1,1]=>1 [1,1,1,1,1,1,2]=>7 [1,1,1,1,1,2,1]=>27 [1,1,1,1,1,3]=>21 [1,1,1,1,2,1,1]=>55 [1,1,1,1,2,2]=>105 [1,1,1,1,3,1]=>85 [1,1,1,1,4]=>35 [1,1,1,2,1,1,1]=>69 [1,1,1,2,1,2]=>203 [1,1,1,2,2,1]=>323 [1,1,1,2,3]=>189 [1,1,1,3,1,1]=>155 [1,1,1,3,2]=>245 [1,1,1,4,1]=>125 [1,1,1,5]=>35 [1,1,2,1,1,1,1]=>55 [1,1,2,1,1,2]=>217 [1,1,2,1,2,1]=>477 [1,1,2,1,3]=>315 [1,1,2,2,1,1]=>449 [1,1,2,2,2]=>791 [1,1,2,3,1]=>531 [1,1,2,4]=>189 [1,1,3,1,1,1]=>155 [1,1,3,1,2]=>413 [1,1,3,2,1]=>573 [1,1,3,3]=>315 [1,1,4,1,1]=>181 [1,1,4,2]=>259 [1,1,5,1]=>99 [1,1,6]=>21 [1,2,1,1,1,1,1]=>27 [1,2,1,1,1,2]=>133 [1,2,1,1,2,1]=>365 [1,2,1,1,3]=>259 [1,2,1,2,1,1]=>477 [1,2,1,2,2]=>875 [1,2,1,3,1]=>643 [1,2,1,4]=>245 [1,2,2,1,1,1]=>323 [1,2,2,1,2]=>917 [1,2,2,2,1]=>1385 [1,2,2,3]=>791 [1,2,3,1,1]=>573 [1,2,3,2]=>875 [1,2,4,1]=>407 [1,2,5]=>105 [1,3,1,1,1,1]=>85 [1,3,1,1,2]=>315 [1,3,1,2,1]=>643 [1,3,1,3]=>413 [1,3,2,1,1]=>531 [1,3,2,2]=>917 [1,3,3,1]=>589 [1,3,4]=>203 [1,4,1,1,1]=>125 [1,4,1,2]=>315 [1,4,2,1]=>407 [1,4,3]=>217 [1,5,1,1]=>99 [1,5,2]=>133 [1,6,1]=>41 [1,7]=>7 [2,1,1,1,1,1,1]=>7 [2,1,1,1,1,2]=>41 [2,1,1,1,2,1]=>133 [2,1,1,1,3]=>99 [2,1,1,2,1,1]=>217 [2,1,1,2,2]=>407 [2,1,1,3,1]=>315 [2,1,1,4]=>125 [2,1,2,1,1,1]=>203 [2,1,2,1,2]=>589 [2,1,2,2,1]=>917 [2,1,2,3]=>531 [2,1,3,1,1]=>413 [2,1,3,2]=>643 [2,1,4,1]=>315 [2,1,5]=>85 [2,2,1,1,1,1]=>105 [2,2,1,1,2]=>407 [2,2,1,2,1]=>875 [2,2,1,3]=>573 [2,2,2,1,1]=>791 [2,2,2,2]=>1385 [2,2,3,1]=>917 [2,2,4]=>323 [2,3,1,1,1]=>245 [2,3,1,2]=>643 [2,3,2,1]=>875 [2,3,3]=>477 [2,4,1,1]=>259 [2,4,2]=>365 [2,5,1]=>133 [2,6]=>27 [3,1,1,1,1,1]=>21 [3,1,1,1,2]=>99 [3,1,1,2,1]=>259 [3,1,1,3]=>181 [3,1,2,1,1]=>315 [3,1,2,2]=>573 [3,1,3,1]=>413 [3,1,4]=>155 [3,2,1,1,1]=>189 [3,2,1,2]=>531 [3,2,2,1]=>791 [3,2,3]=>449 [3,3,1,1]=>315 [3,3,2]=>477 [3,4,1]=>217 [3,5]=>55 [4,1,1,1,1]=>35 [4,1,1,2]=>125 [4,1,2,1]=>245 [4,1,3]=>155 [4,2,1,1]=>189 [4,2,2]=>323 [4,3,1]=>203 [4,4]=>69 [5,1,1,1]=>35 [5,1,2]=>85 [5,2,1]=>105 [5,3]=>55 [6,1,1]=>21 [6,2]=>27 [7,1]=>7 [8]=>1 [1,1,1,1,1,1,1,1,1]=>1 [1,1,1,1,1,1,1,2]=>8 [1,1,1,1,1,1,2,1]=>35 [1,1,1,1,1,1,3]=>28 [1,1,1,1,1,2,1,1]=>83 [1,1,1,1,1,2,2]=>160 [1,1,1,1,1,3,1]=>133 [1,1,1,1,1,4]=>56 [1,1,1,1,2,1,1,1]=>125 [1,1,1,1,2,1,2]=>370 [1,1,1,1,2,2,1]=>595 [1,1,1,1,2,3]=>350 [1,1,1,1,3,1,1]=>295 [1,1,1,1,3,2]=>470 [1,1,1,1,4,1]=>245 [1,1,1,1,5]=>70 [1,1,1,2,1,1,1,1]=>125 [1,1,1,2,1,1,2]=>496 [1,1,1,2,1,2,1]=>1099 [1,1,1,2,1,3]=>728 [1,1,1,2,2,1,1]=>1051 [1,1,1,2,2,2]=>1856 [1,1,1,2,3,1]=>1253 [1,1,1,2,4]=>448 [1,1,1,3,1,1,1]=>379 [1,1,1,3,1,2]=>1016 [1,1,1,3,2,1]=>1421 [1,1,1,3,3]=>784 [1,1,1,4,1,1]=>461 [1,1,1,4,2]=>664 [1,1,1,5,1]=>259 [1,1,1,6]=>56 [1,1,2,1,1,1,1,1]=>83 [1,1,2,1,1,1,2]=>412 [1,1,2,1,1,2,1]=>1141 [1,1,2,1,1,3]=>812 [1,1,2,1,2,1,1]=>1513 [1,1,2,1,2,2]=>2780 [1,1,2,1,3,1]=>2051 [1,1,2,1,4]=>784 [1,1,2,2,1,1,1]=>1051 [1,1,2,2,1,2]=>2990 [1,1,2,2,2,1]=>4529 [1,1,2,2,3]=>2590 [1,1,2,3,1,1]=>1889 [1,1,2,3,2]=>2890 [1,1,2,4,1]=>1351 [1,1,2,5]=>350 [1,1,3,1,1,1,1]=>295 [1,1,3,1,1,2]=>1100 [1,1,3,1,2,1]=>2261 [1,1,3,1,3]=>1456 [1,1,3,2,1,1]=>1889 [1,1,3,2,2]=>3268 [1,1,3,3,1]=>2107 [1,1,3,4]=>728 [1,1,4,1,1,1]=>461 [1,1,4,1,2]=>1168 [1,1,4,2,1]=>1519 [1,1,4,3]=>812 [1,1,5,1,1]=>379 [1,1,5,2]=>512 [1,1,6,1]=>161 [1,1,7]=>28 [1,2,1,1,1,1,1,1]=>35 [1,2,1,1,1,1,2]=>208 [1,2,1,1,1,2,1]=>685 [1,2,1,1,1,3]=>512 [1,2,1,1,2,1,1]=>1141 [1,2,1,1,2,2]=>2144 [1,2,1,1,3,1]=>1667 [1,2,1,1,4]=>664 [1,2,1,2,1,1,1]=>1099 [1,2,1,2,1,2]=>3194 [1,2,1,2,2,1]=>4985 [1,2,1,2,3]=>2890 [1,2,1,3,1,1]=>2261 [1,2,1,3,2]=>3526 [1,2,1,4,1]=>1735 [1,2,1,5]=>470 [1,2,2,1,1,1,1]=>595 [1,2,2,1,1,2]=>2312 [1,2,2,1,2,1]=>4985 [1,2,2,1,3]=>3268 [1,2,2,2,1,1]=>4529 [1,2,2,2,2]=>7936 [1,2,2,3,1]=>5263 [1,2,2,4]=>1856 [1,2,3,1,1,1]=>1421 [1,2,3,1,2]=>3736 [1,2,3,2,1]=>5095 [1,2,3,3]=>2780 [1,2,4,1,1]=>1519 [1,2,4,2]=>2144 [1,2,5,1]=>785 [1,2,6]=>160 [1,3,1,1,1,1,1]=>133 [1,3,1,1,1,2]=>632 [1,3,1,1,2,1]=>1667 [1,3,1,1,3]=>1168 [1,3,1,2,1,1]=>2051 [1,3,1,2,2]=>3736 [1,3,1,3,1]=>2701 [1,3,1,4]=>1016 [1,3,2,1,1,1]=>1253 [1,3,2,1,2]=>3526 [1,3,2,2,1]=>5263 [1,3,2,3]=>2990 [1,3,3,1,1]=>2107 [1,3,3,2]=>3194 [1,3,4,1]=>1457 [1,3,5]=>370 [1,4,1,1,1,1]=>245 [1,4,1,1,2]=>880 [1,4,1,2,1]=>1735 [1,4,1,3]=>1100 [1,4,2,1,1]=>1351 [1,4,2,2]=>2312 [1,4,3,1]=>1457 [1,4,4]=>496 [1,5,1,1,1]=>259 [1,5,1,2]=>632 [1,5,2,1]=>785 [1,5,3]=>412 [1,6,1,1]=>161 [1,6,2]=>208 [1,7,1]=>55 [1,8]=>8 [2,1,1,1,1,1,1,1]=>8 [2,1,1,1,1,1,2]=>55 [2,1,1,1,1,2,1]=>208 [2,1,1,1,1,3]=>161 [2,1,1,1,2,1,1]=>412 [2,1,1,1,2,2]=>785 [2,1,1,1,3,1]=>632 [2,1,1,1,4]=>259 [2,1,1,2,1,1,1]=>496 [2,1,1,2,1,2]=>1457 [2,1,1,2,2,1]=>2312 [2,1,1,2,3]=>1351 [2,1,1,3,1,1]=>1100 [2,1,1,3,2]=>1735 [2,1,1,4,1]=>880 [2,1,1,5]=>245 [2,1,2,1,1,1,1]=>370 [2,1,2,1,1,2]=>1457 [2,1,2,1,2,1]=>3194 [2,1,2,1,3]=>2107 [2,1,2,2,1,1]=>2990 [2,1,2,2,2]=>5263 [2,1,2,3,1]=>3526 [2,1,2,4]=>1253 [2,1,3,1,1,1]=>1016 [2,1,3,1,2]=>2701 [2,1,3,2,1]=>3736 [2,1,3,3]=>2051 [2,1,4,1,1]=>1168 [2,1,4,2]=>1667 [2,1,5,1]=>632 [2,1,6]=>133 [2,2,1,1,1,1,1]=>160 [2,2,1,1,1,2]=>785 [2,2,1,1,2,1]=>2144 [2,2,1,1,3]=>1519 [2,2,1,2,1,1]=>2780 [2,2,1,2,2]=>5095 [2,2,1,3,1]=>3736 [2,2,1,4]=>1421 [2,2,2,1,1,1]=>1856 [2,2,2,1,2]=>5263 [2,2,2,2,1]=>7936 [2,2,2,3]=>4529 [2,2,3,1,1]=>3268 [2,2,3,2]=>4985 [2,2,4,1]=>2312 [2,2,5]=>595 [2,3,1,1,1,1]=>470 [2,3,1,1,2]=>1735 [2,3,1,2,1]=>3526 [2,3,1,3]=>2261 [2,3,2,1,1]=>2890 [2,3,2,2]=>4985 [2,3,3,1]=>3194 [2,3,4]=>1099 [2,4,1,1,1]=>664 [2,4,1,2]=>1667 [2,4,2,1]=>2144 [2,4,3]=>1141 [2,5,1,1]=>512 [2,5,2]=>685 [2,6,1]=>208 [2,7]=>35 [3,1,1,1,1,1,1]=>28 [3,1,1,1,1,2]=>161 [3,1,1,1,2,1]=>512 [3,1,1,1,3]=>379 [3,1,1,2,1,1]=>812 [3,1,1,2,2]=>1519 [3,1,1,3,1]=>1168 [3,1,1,4]=>461 [3,1,2,1,1,1]=>728 [3,1,2,1,2]=>2107 [3,1,2,2,1]=>3268 [3,1,2,3]=>1889 [3,1,3,1,1]=>1456 [3,1,3,2]=>2261 [3,1,4,1]=>1100 [3,1,5]=>295 [3,2,1,1,1,1]=>350 [3,2,1,1,2]=>1351 [3,2,1,2,1]=>2890 [3,2,1,3]=>1889 [3,2,2,1,1]=>2590 [3,2,2,2]=>4529 [3,2,3,1]=>2990 [3,2,4]=>1051 [3,3,1,1,1]=>784 [3,3,1,2]=>2051 [3,3,2,1]=>2780 [3,3,3]=>1513 [3,4,1,1]=>812 [3,4,2]=>1141 [3,5,1]=>412 [3,6]=>83 [4,1,1,1,1,1]=>56 [4,1,1,1,2]=>259 [4,1,1,2,1]=>664 [4,1,1,3]=>461 [4,1,2,1,1]=>784 [4,1,2,2]=>1421 [4,1,3,1]=>1016 [4,1,4]=>379 [4,2,1,1,1]=>448 [4,2,1,2]=>1253 [4,2,2,1]=>1856 [4,2,3]=>1051 [4,3,1,1]=>728 [4,3,2]=>1099 [4,4,1]=>496 [4,5]=>125 [5,1,1,1,1]=>70 [5,1,1,2]=>245 [5,1,2,1]=>470 [5,1,3]=>295 [5,2,1,1]=>350 [5,2,2]=>595 [5,3,1]=>370 [5,4]=>125 [6,1,1,1]=>56 [6,1,2]=>133 [6,2,1]=>160 [6,3]=>83 [7,1,1]=>28 [7,2]=>35 [8,1]=>8 [9]=>1 [1,1,1,1,1,1,1,1,1,1]=>1 [1,1,1,1,1,1,2,2]=>231 [1,1,1,1,2,1,1,2]=>999 [1,1,1,1,2,2,1,1]=>2149 [1,1,1,1,3,3]=>1674 [1,1,1,1,5,1]=>574 [1,1,2,1,1,1,1,2]=>711 [1,1,2,1,1,2,1,1]=>3961 [1,1,2,1,2,3]=>10206 [1,1,2,2,1,1,1,1]=>2149 [1,1,2,2,2,2]=>28839 [1,1,3,1,1,3]=>4536 [1,1,3,2,1,2]=>13941 [1,1,3,3,1,1]=>8371 [1,1,4,4]=>2064 [1,1,7,1]=>244 [1,2,1,1,4,1]=>5191 [1,2,6,1]=>1369 [1,3,1,1,3,1]=>8371 [1,3,5,1]=>3079 [1,4,1,1,2,1]=>5191 [1,4,4,1]=>3961 [1,5,1,1,1,1]=>574 [1,5,3,1]=>3079 [1,6,1,2]=>1134 [1,6,2,1]=>1369 [1,7,1,1]=>244 [1,8,1]=>71 [1,9]=>9 [2,1,1,1,1,1,1,2]=>71 [2,1,1,1,1,2,1,1]=>711 [2,1,1,1,2,3]=>2906 [2,1,1,2,1,1,1,1]=>999 [2,1,1,2,2,2]=>14759 [2,1,2,1,1,3]=>6056 [2,1,2,2,1,2]=>22121 [2,1,2,3,1,1]=>13941 [2,1,3,4]=>5264 [2,1,6,1]=>1134 [2,2,1,1,1,1,1,1]=>231 [2,2,1,1,2,2]=>13991 [2,2,2,1,1,2]=>14759 [2,2,2,2,1,1]=>28839 [2,2,3,3]=>17594 [3,1,1,1,1,3]=>701 [3,1,1,2,1,2]=>6056 [3,1,1,3,1,1]=>4536 [3,1,2,4]=>4949 [3,2,1,1,1,2]=>2906 [3,2,1,2,1,1]=>10206 [3,2,2,3]=>16451 [3,3,1,1,1,1]=>1674 [3,3,2,2]=>17594 [4,1,1,4]=>1301 [4,2,1,3]=>4949 [4,3,1,2]=>5264 [4,4,1,1]=>2064 [5,5]=>251 [8,1,1]=>36 [9,1]=>9 [10,1]=>10 [1,10]=>10 [1,8,1,1]=>351 [1,7,2,1]=>2221 [1,6,3,1]=>5851 [1,5,4,1]=>9151 [1,4,5,1]=>9151 [1,3,6,1]=>5851 [1,2,7,1]=>2221 [1,1,8,1]=>351 [5,5,1,1]=>11825 [1,1,5,5]=>11825 [1,1,4,1,1,4]=>55220 [4,1,1,4,1,1]=>55220 [1,1,3,1,2,4]=>185108 [1,1,4,2,1,3]=>215160 [3,1,2,4,1,1]=>215160 [4,2,1,3,1,1]=>185108 [1,1,2,1,3,4]=>150227 [1,1,3,1,1,1,1,3]=>25080 [1,1,4,3,1,2]=>230241 [2,2,4,4]=>157475 [2,1,3,4,1,1]=>230241 [3,1,1,1,1,3,1,1]=>25080 [4,4,2,2]=>157475 [4,3,1,2,1,1]=>150227 [1,1,1,1,4,4]=>16995 [1,1,2,1,2,1,1,3]=>169455 [1,1,3,2,2,3]=>635745 [1,1,3,1,1,2,1,2]=>221529 [1,1,4,4,1,1]=>90509 [2,2,3,1,1,3]=>385395 [2,1,2,1,1,3,1,1]=>221529 [3,3,3,3]=>315523 [3,2,2,3,1,1]=>635745 [3,1,1,3,2,2]=>385395 [3,1,1,2,1,2,1,1]=>169455 [4,4,1,1,1,1]=>16995 [1,1,1,1,3,1,1,3]=>34375 [1,1,2,1,1,1,2,3]=>75570 [1,1,2,1,2,2,1,2]=>622314 [1,1,3,2,1,1,1,2]=>111474 [1,1,3,1,1,3,1,1]=>166541 [2,2,2,1,2,3]=>996390 [2,2,3,2,1,2]=>1152546 [2,1,1,2,3,3]=>280774 [2,1,1,1,2,3,1,1]=>111474 [2,1,2,3,2,2]=>1152546 [2,1,2,2,1,2,1,1]=>622314 [3,3,2,1,1,2]=>280774 [3,2,1,2,2,2]=>996390 [3,2,1,1,1,2,1,1]=>75570 [3,1,1,3,1,1,1,1]=>34375 [1,1,2,1,1,1,1,1,1,2]=>1749 [2,2,1,1,1,1,1,1,1,1]=>429 [1,1,2,2,3,3]=>538890 [1,1,1,1,1,1,2,1,1,2]=>3157 [1,1,1,1,1,1,1,1,2,2]=>429 [2,2,1,1,2,1,1,2]=>199341 [2,2,1,1,1,1,2,2]=>66989 [2,1,1,2,2,1,1,2]=>228205 [1,1,2,2,2,1,1,2]=>450021 [2,1,1,2,1,1,2,2]=>199341 [1,1,2,2,1,1,2,2]=>421421 [1,1,1,1,3,2,1,2]=>107866 [1,1,2,1,1,2,2,2]=>396077 [3,3,1,1,2,2]=>312102 [1,1,1,1,2,1,1,1,1,2]=>4741 [1,1,1,1,2,2,2,2]=>203181 [3,2,2,2,1,2]=>1022845 [2,1,1,1,2,2,1,2]=>174514 [1,1,1,1,2,1,2,3]=>69850 [3,2,1,1,1,1,1,2]=>9910 [2,1,1,1,1,1,1,1,1,2]=>109 [2,1,1,1,1,2,2,2]=>72621 [2,2,2,1,1,1,1,2]=>72621 [2,2,2,2,2,2]=>2702765 [1,1,1,1,1,1,3,3]=>5698 [3,1,2,3,1,2]=>527284 [2,1,3,3,1,2]=>578665 [3,1,1,1,1,2,1,2]=>30700 [2,1,2,1,1,2,1,2]=>291169 [3,1,1,2,1,1,1,2]=>45715 [2,1,2,2,1,1,1,2]=>174514 [1,1,3,3,2,2]=>686906 [2,2,1,1,3,3]=>312102 [3,3,1,1,1,1,1,1]=>5698 [2,1,2,3,1,1,1,1]=>107866 [1,1,3,3,1,1,1,1]=>65154 [2,2,3,3,1,1]=>686906 [1,1,1,1,3,3,1,1]=>65154 [3,1,1,1,1,1,1,3]=>1891 [3,1,1,2,2,3]=>294106 [3,2,1,1,2,3]=>251371 [2,1,1,1,1,1,2,3]=>9910 [1,1,2,1,2,3,1,1]=>392822 [4,2,1,1,1,3]=>43251 [3,2,2,1,1,3]=>294106 [4,2,1,2,1,2]=>262932 [2,1,2,2,2,3]=>1022845 [4,3,1,1,1,2]=>46947 [3,3,2,2,1,1]=>538890 [1,1,2,2,2,2,1,1]=>880529 [2,2,1,1,2,2,1,1]=>421421 [2,1,1,1,2,1,1,3]=>45715 [2,2,2,2,1,1,1,1]=>203181 [2,1,1,2,1,1,1,1,1,1]=>3157 [2,1,1,1,3,4]=>46947 [2,1,1,1,1,1,1,2,1,1]=>1749 [2,1,2,1,1,1,1,3]=>30700 [2,2,2,1,1,2,1,1]=>396077 [2,1,1,1,1,2,1,1,1,1]=>4741 [2,1,2,1,2,4]=>262932 [2,1,1,2,2,2,1,1]=>450021 [2,1,3,2,1,3]=>527284 [2,1,3,1,1,4]=>123540 [3,1,1,1,2,4]=>43251 [1,1,2,1,1,2,1,1,1,1]=>26709 [1,1,1,1,2,1,1,2,1,1]=>26709 [1,1,2,2,1,1,1,1,1,1]=>6909 [1,1,1,1,2,2,1,1,1,1]=>15049 [1,1,1,1,1,1,2,2,1,1]=>6909 [1,1,1,1,1,1,1,1,1,1,1,1]=>1 [5,4,1,2]=>39225 [4,1,1,3,1,2]=>123540 [4,3,2,3]=>215346 [5,3,1,3]=>54966 [4,1,1,2,1,3]=>90771 [3,2,1,2,1,1,1,1]=>69850 [1,1,3,2,1,2,1,1]=>392822 [5,2,1,4]=>36926 [5,1,1,5]=>8051 [3,1,2,2,1,3]=>458083 [4,2,2,4]=>147443 [3,1,2,1,1,4]=>90771 [4,1,1,1,1,4]=>8051 [3,1,3,5]=>54966 [6,6]=>923 [4,1,2,5]=>36926 [2,1,4,5]=>39225 [3,2,3,4]=>215346 [1,1,2,1,1,1,1,2,1,1]=>18041
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Description
The number of ribbon shaped standard tableaux.
A ribbon is a connected skew shape which does not contain a $2\times 2$ square. The set of ribbon shapes are therefore in bijection with integer compositons, the parts of the composition specify the row lengths. This statistic records the number of standard tableaux of the given shape.
This is also the size of the preimage of the map 'descent composition' Mp00071descent composition from permutations to integer compositions: reading a tableau from bottom to top we obtain a permutation whose descent set is as prescribed.
Code
def composition_to_ribbon(c):
    inner = []
    outer = []
    indent = 0
    for p in reversed(c):
        if indent > 0:
            inner.append(indent)
        outer.append(p+indent)
        indent += p-1
    return SkewPartition([outer[::-1], inner[::-1]])

def statistic(c):
    return StandardSkewTableaux(composition_to_ribbon(c)).cardinality()

# alternative implementation

def descents_composition(elt):
    if len(elt) == 0:
        return Composition([])
    d = [-1] + elt.descents() + [len(elt)-1]
    return Composition([ d[i+1]-d[i] for i in range(len(d)-1)])

@cached_function
def preimages(level):
    result = dict()
    for el in Permutations(level):
        image = descents_composition(el)
        result[image] = result.get(image, 0) + 1
    return result

def statistic(x):
    return preimages(x.size()).get(x, 0)

Created
Sep 11, 2015 at 21:12 by Martin Rubey
Updated
Sep 16, 2019 at 09:44 by Martin Rubey