The number of 1...d-avoiding permutations of length d+r for SYMBOLIC d but numeric r (between 1 and 30)

By Shalosh B. EKHAD and Doron ZEILBERGER

Dedicated to Ira Martin Gessel (b. April 9, 1951), on his millionth_{2}

birthday

Preface: How many permutations are there of length googol+30 avoiding an increasing subsequence of length googol?

This number is way too big for our physical universe, but the number of permutations of length googol+30 that contains at least one increasing subsequence of length googol is

3769987628815905643852921525646105664146833823621994801456991357113502936781270538054719048039675278 0769193354371721350001524610578097700045972792823897290959624203896101981952929640805170129282073883 4740018807571147534091229951249435913149171795302592312477456091277812321956212802204785578598020255 5625008802850838455586257402947256848380647181479993222566420025908679106917004348077812428261510240 6340176300585397517990032393036653951304924586489968650809789292291489270968710994809677050176596751 0725956202350750841376095024046396844968511243494784162014881795337835528626142808150073111101283361 0980701571937952824136796425017224636196853995950587943259043687431653922927840572864396105085190223

2582799067810378389890635196322425667467335158890500820128768331750855469963050322432973194472331947 0989825934469696079344723053679001130033667827524966034661782064851068214182454731365743413486729730 0631055444127725930013792836515384850702346797298406803049230145697433567004811555984158378611125895 0145768901348725550726037527669812626353266837685037397408862767082018239579392663024131792105407280 4788720840618514463465035392103884394981202007834724110094447116613439128758285044269471808502083275 6629374247928521501786839409853287740758570056230853738462527374534709641735458487560816949365616486 0695626913029699922648102091615529949414940648588048836485372758775808743231365617459515329190972398 7074543946415578728439994306071279654045146432379557541358408978156863172980419720839292761025261752 6805876626590163265795592248178664681630980893821587688413815206609216082514983787883386977226071420 21649147728993592578961422177700294482596740993986519357246959930668146505085270714475561150113747221