From: David W. Wilson (wilson(AT)aprisma.com) Date: Fri, 07 Sep 2001 15:59:49 -0400 Extended tables for A008304 and A064315. EIS sequence A008304 f(n,k) = number of permutations of {1,...,n} whose longest run of increasing adjacent elements is of length k. f(1,1) = 1 f(2,1) = 1 f(2,2) = 1 f(3,1) = 1 f(3,2) = 4 f(3,3) = 1 f(4,1) = 1 f(4,2) = 16 f(4,3) = 6 f(4,4) = 1 f(5,1) = 1 f(5,2) = 69 f(5,3) = 41 f(5,4) = 8 f(5,5) = 1 f(6,1) = 1 f(6,2) = 348 f(6,3) = 293 f(6,4) = 67 f(6,5) = 10 f(6,6) = 1 f(7,1) = 1 f(7,2) = 2016 f(7,3) = 2309 f(7,4) = 602 f(7,5) = 99 f(7,6) = 12 f(7,7) = 1 f(8,1) = 1 f(8,2) = 13357 f(8,3) = 19975 f(8,4) = 5811 f(8,5) = 1024 f(8,6) = 137 f(8,7) = 14 f(8,8) = 1 f(9,1) = 1 f(9,2) = 99376 f(9,3) = 189524 f(9,4) = 60875 f(9,5) = 11304 f(9,6) = 1602 f(9,7) = 181 f(9,8) = 16 f(9,9) = 1 f(10,1) = 1 f(10,2) = 822040 f(10,3) = 1960041 f(10,4) = 690729 f(10,5) = 133669 f(10,6) = 19710 f(10,7) = 2360 f(10,8) = 231 f(10,9) = 18 f(10,10) = 1 f(11,1) = 1 f(11,2) = 7477161 f(11,3) = 21993884 f(11,4) = 8457285 f(11,5) = 1695429 f(11,6) = 257400 f(11,7) = 32010 f(11,8) = 3322 f(11,9) = 287 f(11,10) = 20 f(11,11) = 1 f(12,1) = 1 f(12,2) = 74207208 f(12,3) = 266361634 f(12,4) = 111323149 f(12,5) = 23023811 f(12,6) = 3574957 f(12,7) = 456720 f(12,8) = 49236 f(12,9) = 4512 f(12,10) = 349 f(12,11) = 22 f(12,12) = 1 f(13,1) = 1 f(13,2) = 797771520 f(13,3) = 3465832370 f(13,4) = 1569068565 f(13,5) = 333840443 f(13,6) = 52785901 f(13,7) = 6881160 f(13,8) = 761904 f(13,9) = 72540 f(13,10) = 5954 f(13,11) = 417 f(13,12) = 24 f(13,13) = 1 f(14,1) = 1 f(14,2) = 9236662345 f(14,3) = 48245601976 f(14,4) = 23592426102 f(14,5) = 5153118154 f(14,6) = 827242933 f(14,7) = 109546009 f(14,8) = 12372360 f(14,9) = 1209936 f(14,10) = 103194 f(14,11) = 7672 f(14,12) = 491 f(14,13) = 26 f(14,14) = 1 f(15,1) = 1 f(15,2) = 114579019468 f(15,3) = 715756932697 f(15,4) = 377105857043 f(15,5) = 84426592621 f(15,6) = 13730434111 f(15,7) = 1841298059 f(15,8) = 211170960 f(15,9) = 21064680 f(15,10) = 1845480 f(15,11) = 142590 f(15,12) = 9690 f(15,13) = 571 f(15,14) = 28 f(15,15) = 1 f(16,1) = 1 f(16,2) = 1516103040832 f(16,3) = 11277786883720 f(16,4) = 6387313185576 f(16,5) = 1463941342191 f(16,6) = 240806565782 f(16,7) = 32629877967 f(16,8) = 3788091451 f(16,9) = 383685120 f(16,10) = 34288800 f(16,11) = 2721600 f(16,12) = 192240 f(16,13) = 12032 f(16,14) = 657 f(16,15) = 30 f(16,16) = 1 f(17,1) = 1 f(17,2) = 21314681315997 f(17,3) = 188135296651083 f(17,4) = 114303481217657 f(17,5) = 26793750988542 f(17,6) = 4452251786946 f(17,7) = 608572228291 f(17,8) = 71356438043 f(17,9) = 7315701120 f(17,10) = 663848640 f(17,11) = 53749920 f(17,12) = 3900480 f(17,13) = 253776 f(17,14) = 14722 f(17,15) = 749 f(17,16) = 32 f(17,17) = 1 f(18,1) = 1 f(18,2) = 317288088082404 f(18,3) = 3313338641692957 f(18,4) = 2155348564847332 f(18,5) = 516319125748337 f(18,6) = 86585391630673 f(18,7) = 11923667699474 f(18,8) = 1409672722481 f(18,9) = 145957544981 f(18,10) = 13406178240 f(18,11) = 1102187520 f(18,12) = 81591840 f(18,13) = 5454144 f(18,14) = 328950 f(18,15) = 17784 f(18,16) = 847 f(18,17) = 34 f(18,18) = 1 f(19,1) = 1 f(19,2) = 4985505271920096 f(19,3) = 61444453534759589 f(19,4) = 42719058006864690 f(19,5) = 10451197169218523 f(19,6) = 1767406549387381 f(19,7) = 244964063143590 f(19,8) = 29163603260677 f(19,9) = 3044416187213 f(19,10) = 282398538240 f(19,11) = 23500653120 f(19,12) = 1766525760 f(19,13) = 120466080 f(19,14) = 7465176 f(19,15) = 419634 f(19,16) = 21242 f(19,17) = 951 f(19,18) = 36 f(19,19) = 1 f(20,1) = 1 f(20,2) = 82459612672301845 f(20,3) = 1196988740015236617 f(20,4) = 887953677898186108 f(20,5) = 221738082618710329 f(20,6) = 37790452850585180 f(20,7) = 5267496652725480 f(20,8) = 630867328411136 f(20,9) = 66312765615259 f(20,10) = 6201593613645 f(20,11) = 521194867200 f(20,12) = 39656131200 f(20,13) = 2746068480 f(20,14) = 173606040 f(20,15) = 10027440 f(20,16) = 527820 f(20,17) = 25120 f(20,18) = 1061 f(20,19) = 38 f(20,20) = 1 EIS sequence A064315 f(n,k) = number of permutations of {1,...,n} whose shortest run of increasing adjacent elements is of length k. f(1,1) = 1 f(2,1) = 1 f(2,2) = 1 f(3,1) = 5 f(3,2) = 0 f(3,3) = 1 f(4,1) = 18 f(4,2) = 5 f(4,3) = 0 f(4,4) = 1 f(5,1) = 101 f(5,2) = 18 f(5,3) = 0 f(5,4) = 0 f(5,5) = 1 f(6,1) = 611 f(6,2) = 89 f(6,3) = 19 f(6,4) = 0 f(6,5) = 0 f(6,6) = 1 f(7,1) = 4452 f(7,2) = 519 f(7,3) = 68 f(7,4) = 0 f(7,5) = 0 f(7,6) = 0 f(7,7) = 1 f(8,1) = 36287 f(8,2) = 3853 f(8,3) = 110 f(8,4) = 69 f(8,5) = 0 f(8,6) = 0 f(8,7) = 0 f(8,8) = 1 f(9,1) = 333395 f(9,2) = 27555 f(9,3) = 1679 f(9,4) = 250 f(9,5) = 0 f(9,6) = 0 f(9,7) = 0 f(9,8) = 0 f(9,9) = 1 f(10,1) = 3382758 f(10,2) = 233431 f(10,3) = 11941 f(10,4) = 418 f(10,5) = 251 f(10,6) = 0 f(10,7) = 0 f(10,8) = 0 f(10,9) = 0 f(10,10) = 1 f(11,1) = 37688597 f(11,2) = 2167152 f(11,3) = 59470 f(11,4) = 658 f(11,5) = 922 f(11,6) = 0 f(11,7) = 0 f(11,8) = 0 f(11,9) = 0 f(11,10) = 0 f(11,11) = 1 f(12,1) = 456839351 f(12,2) = 21596120 f(12,3) = 528974 f(12,4) = 34649 f(12,5) = 1582 f(12,6) = 923 f(12,7) = 0 f(12,8) = 0 f(12,9) = 0 f(12,10) = 0 f(12,11) = 0 f(12,12) = 1 f(13,1) = 5989023768 f(13,2) = 232817282 f(13,3) = 4907480 f(13,4) = 266267 f(13,5) = 2572 f(13,6) = 3430 f(13,7) = 0 f(13,8) = 0 f(13,9) = 0 f(13,10) = 0 f(13,11) = 0 f(13,12) = 0 f(13,13) = 1 f(14,1) = 84421235807 f(14,2) = 2718706924 f(14,3) = 36965659 f(14,4) = 1369372 f(14,5) = 4002 f(14,6) = 6004 f(14,7) = 3431 f(14,8) = 0 f(14,9) = 0 f(14,10) = 0 f(14,11) = 0 f(14,12) = 0 f(14,13) = 0 f(14,14) = 1 f(15,1) = 1273482972215 f(15,2) = 33814848445 f(15,3) = 370685663 f(15,4) = 5082045 f(15,5) = 756755 f(15,6) = 10008 f(15,7) = 12868 f(15,8) = 0 f(15,9) = 0 f(15,10) = 0 f(15,11) = 0 f(15,12) = 0 f(15,13) = 0 f(15,14) = 0 f(15,15) = 1 f(16,1) = 20470309460322 f(16,2) = 448311181346 f(16,3) = 4086527638 f(16,4) = 76637635 f(16,5) = 6029297 f(16,6) = 16014 f(16,7) = 22878 f(16,8) = 12869 f(16,9) = 0 f(16,10) = 0 f(16,11) = 0 f(16,12) = 0 f(16,13) = 0 f(16,14) = 0 f(16,15) = 0 f(16,16) = 1 f(17,1) = 349326503482301 f(17,2) = 6319365554730 f(17,3) = 40650345653 f(17,4) = 876858377 f(17,5) = 31742676 f(17,6) = 24750 f(17,7) = 38894 f(17,8) = 48618 f(17,9) = 0 f(17,10) = 0 f(17,11) = 0 f(17,12) = 0 f(17,13) = 0 f(17,14) = 0 f(17,15) = 0 f(17,16) = 0 f(17,17) = 1 f(18,1) = 6307682420743595 f(18,2) = 94225534689624 f(18,3) = 458464525689 f(18,4) = 7147459470 f(18,5) = 120956707 f(18,6) = 17153135 f(18,7) = 63646 f(18,8) = 87514 f(18,9) = 48619 f(18,10) = 0 f(18,11) = 0 f(18,12) = 0 f(18,13) = 0 f(18,14) = 0 f(18,15) = 0 f(18,16) = 0 f(18,17) = 0 f(18,18) = 1 f(19,1) = 120157254334350828 f(19,2) = 1481940898130323 f(19,3) = 5857242113368 f(19,4) = 47396097511 f(19,5) = 398182782 f(19,6) = 139520497 f(19,7) = 100774 f(19,8) = 151162 f(19,9) = 184754 f(19,10) = 0 f(19,11) = 0 f(19,12) = 0 f(19,13) = 0 f(19,14) = 0 f(19,15) = 0 f(19,16) = 0 f(19,17) = 0 f(19,18) = 0 f(19,19) = 1 f(20,1) = 2408293016265606623 f(20,2) = 24536143182460549 f(20,3) = 72283039099029 f(20,4) = 552146011437 f(20,5) = 12794989968 f(20,6) = 747544744 f(20,7) = 155038 f(20,8) = 251938 f(20,9) = 335918 f(20,10) = 184755 f(20,11) = 0 f(20,12) = 0 f(20,13) = 0 f(20,14) = 0 f(20,15) = 0 f(20,16) = 0 f(20,17) = 0 f(20,18) = 0 f(20,19) = 0 f(20,20) = 1